YES

We show the termination of the TRS R:

  active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
  active(sqr(|0|())) -> mark(|0|())
  active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
  active(dbl(|0|())) -> mark(|0|())
  active(dbl(s(X))) -> mark(s(s(dbl(X))))
  active(add(|0|(),X)) -> mark(X)
  active(add(s(X),Y)) -> mark(s(add(X,Y)))
  active(first(|0|(),X)) -> mark(nil())
  active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
  mark(terms(X)) -> active(terms(mark(X)))
  mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
  mark(recip(X)) -> active(recip(mark(X)))
  mark(sqr(X)) -> active(sqr(mark(X)))
  mark(s(X)) -> active(s(X))
  mark(|0|()) -> active(|0|())
  mark(add(X1,X2)) -> active(add(mark(X1),mark(X2)))
  mark(dbl(X)) -> active(dbl(mark(X)))
  mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
  mark(nil()) -> active(nil())
  terms(mark(X)) -> terms(X)
  terms(active(X)) -> terms(X)
  cons(mark(X1),X2) -> cons(X1,X2)
  cons(X1,mark(X2)) -> cons(X1,X2)
  cons(active(X1),X2) -> cons(X1,X2)
  cons(X1,active(X2)) -> cons(X1,X2)
  recip(mark(X)) -> recip(X)
  recip(active(X)) -> recip(X)
  sqr(mark(X)) -> sqr(X)
  sqr(active(X)) -> sqr(X)
  s(mark(X)) -> s(X)
  s(active(X)) -> s(X)
  add(mark(X1),X2) -> add(X1,X2)
  add(X1,mark(X2)) -> add(X1,X2)
  add(active(X1),X2) -> add(X1,X2)
  add(X1,active(X2)) -> add(X1,X2)
  dbl(mark(X)) -> dbl(X)
  dbl(active(X)) -> dbl(X)
  first(mark(X1),X2) -> first(X1,X2)
  first(X1,mark(X2)) -> first(X1,X2)
  first(active(X1),X2) -> first(X1,X2)
  first(X1,active(X2)) -> first(X1,X2)

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: active#(terms(N)) -> mark#(cons(recip(sqr(N)),terms(s(N))))
p2: active#(terms(N)) -> cons#(recip(sqr(N)),terms(s(N)))
p3: active#(terms(N)) -> recip#(sqr(N))
p4: active#(terms(N)) -> sqr#(N)
p5: active#(terms(N)) -> terms#(s(N))
p6: active#(terms(N)) -> s#(N)
p7: active#(sqr(|0|())) -> mark#(|0|())
p8: active#(sqr(s(X))) -> mark#(s(add(sqr(X),dbl(X))))
p9: active#(sqr(s(X))) -> s#(add(sqr(X),dbl(X)))
p10: active#(sqr(s(X))) -> add#(sqr(X),dbl(X))
p11: active#(sqr(s(X))) -> sqr#(X)
p12: active#(sqr(s(X))) -> dbl#(X)
p13: active#(dbl(|0|())) -> mark#(|0|())
p14: active#(dbl(s(X))) -> mark#(s(s(dbl(X))))
p15: active#(dbl(s(X))) -> s#(s(dbl(X)))
p16: active#(dbl(s(X))) -> s#(dbl(X))
p17: active#(dbl(s(X))) -> dbl#(X)
p18: active#(add(|0|(),X)) -> mark#(X)
p19: active#(add(s(X),Y)) -> mark#(s(add(X,Y)))
p20: active#(add(s(X),Y)) -> s#(add(X,Y))
p21: active#(add(s(X),Y)) -> add#(X,Y)
p22: active#(first(|0|(),X)) -> mark#(nil())
p23: active#(first(s(X),cons(Y,Z))) -> mark#(cons(Y,first(X,Z)))
p24: active#(first(s(X),cons(Y,Z))) -> cons#(Y,first(X,Z))
p25: active#(first(s(X),cons(Y,Z))) -> first#(X,Z)
p26: mark#(terms(X)) -> active#(terms(mark(X)))
p27: mark#(terms(X)) -> terms#(mark(X))
p28: mark#(terms(X)) -> mark#(X)
p29: mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2))
p30: mark#(cons(X1,X2)) -> cons#(mark(X1),X2)
p31: mark#(cons(X1,X2)) -> mark#(X1)
p32: mark#(recip(X)) -> active#(recip(mark(X)))
p33: mark#(recip(X)) -> recip#(mark(X))
p34: mark#(recip(X)) -> mark#(X)
p35: mark#(sqr(X)) -> active#(sqr(mark(X)))
p36: mark#(sqr(X)) -> sqr#(mark(X))
p37: mark#(sqr(X)) -> mark#(X)
p38: mark#(s(X)) -> active#(s(X))
p39: mark#(|0|()) -> active#(|0|())
p40: mark#(add(X1,X2)) -> active#(add(mark(X1),mark(X2)))
p41: mark#(add(X1,X2)) -> add#(mark(X1),mark(X2))
p42: mark#(add(X1,X2)) -> mark#(X1)
p43: mark#(add(X1,X2)) -> mark#(X2)
p44: mark#(dbl(X)) -> active#(dbl(mark(X)))
p45: mark#(dbl(X)) -> dbl#(mark(X))
p46: mark#(dbl(X)) -> mark#(X)
p47: mark#(first(X1,X2)) -> active#(first(mark(X1),mark(X2)))
p48: mark#(first(X1,X2)) -> first#(mark(X1),mark(X2))
p49: mark#(first(X1,X2)) -> mark#(X1)
p50: mark#(first(X1,X2)) -> mark#(X2)
p51: mark#(nil()) -> active#(nil())
p52: terms#(mark(X)) -> terms#(X)
p53: terms#(active(X)) -> terms#(X)
p54: cons#(mark(X1),X2) -> cons#(X1,X2)
p55: cons#(X1,mark(X2)) -> cons#(X1,X2)
p56: cons#(active(X1),X2) -> cons#(X1,X2)
p57: cons#(X1,active(X2)) -> cons#(X1,X2)
p58: recip#(mark(X)) -> recip#(X)
p59: recip#(active(X)) -> recip#(X)
p60: sqr#(mark(X)) -> sqr#(X)
p61: sqr#(active(X)) -> sqr#(X)
p62: s#(mark(X)) -> s#(X)
p63: s#(active(X)) -> s#(X)
p64: add#(mark(X1),X2) -> add#(X1,X2)
p65: add#(X1,mark(X2)) -> add#(X1,X2)
p66: add#(active(X1),X2) -> add#(X1,X2)
p67: add#(X1,active(X2)) -> add#(X1,X2)
p68: dbl#(mark(X)) -> dbl#(X)
p69: dbl#(active(X)) -> dbl#(X)
p70: first#(mark(X1),X2) -> first#(X1,X2)
p71: first#(X1,mark(X2)) -> first#(X1,X2)
p72: first#(active(X1),X2) -> first#(X1,X2)
p73: first#(X1,active(X2)) -> first#(X1,X2)

and R consists of:

r1: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
r2: active(sqr(|0|())) -> mark(|0|())
r3: active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
r4: active(dbl(|0|())) -> mark(|0|())
r5: active(dbl(s(X))) -> mark(s(s(dbl(X))))
r6: active(add(|0|(),X)) -> mark(X)
r7: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r8: active(first(|0|(),X)) -> mark(nil())
r9: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r10: mark(terms(X)) -> active(terms(mark(X)))
r11: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r12: mark(recip(X)) -> active(recip(mark(X)))
r13: mark(sqr(X)) -> active(sqr(mark(X)))
r14: mark(s(X)) -> active(s(X))
r15: mark(|0|()) -> active(|0|())
r16: mark(add(X1,X2)) -> active(add(mark(X1),mark(X2)))
r17: mark(dbl(X)) -> active(dbl(mark(X)))
r18: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r19: mark(nil()) -> active(nil())
r20: terms(mark(X)) -> terms(X)
r21: terms(active(X)) -> terms(X)
r22: cons(mark(X1),X2) -> cons(X1,X2)
r23: cons(X1,mark(X2)) -> cons(X1,X2)
r24: cons(active(X1),X2) -> cons(X1,X2)
r25: cons(X1,active(X2)) -> cons(X1,X2)
r26: recip(mark(X)) -> recip(X)
r27: recip(active(X)) -> recip(X)
r28: sqr(mark(X)) -> sqr(X)
r29: sqr(active(X)) -> sqr(X)
r30: s(mark(X)) -> s(X)
r31: s(active(X)) -> s(X)
r32: add(mark(X1),X2) -> add(X1,X2)
r33: add(X1,mark(X2)) -> add(X1,X2)
r34: add(active(X1),X2) -> add(X1,X2)
r35: add(X1,active(X2)) -> add(X1,X2)
r36: dbl(mark(X)) -> dbl(X)
r37: dbl(active(X)) -> dbl(X)
r38: first(mark(X1),X2) -> first(X1,X2)
r39: first(X1,mark(X2)) -> first(X1,X2)
r40: first(active(X1),X2) -> first(X1,X2)
r41: first(X1,active(X2)) -> first(X1,X2)

The estimated dependency graph contains the following SCCs:

  {p1, p8, p14, p18, p19, p23, p26, p28, p29, p31, p32, p34, p35, p37, p38, p40, p42, p43, p44, p46, p47, p49, p50}
  {p54, p55, p56, p57}
  {p58, p59}
  {p60, p61}
  {p52, p53}
  {p62, p63}
  {p64, p65, p66, p67}
  {p68, p69}
  {p70, p71, p72, p73}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: active#(terms(N)) -> mark#(cons(recip(sqr(N)),terms(s(N))))
p2: mark#(first(X1,X2)) -> mark#(X2)
p3: mark#(first(X1,X2)) -> mark#(X1)
p4: mark#(first(X1,X2)) -> active#(first(mark(X1),mark(X2)))
p5: active#(first(s(X),cons(Y,Z))) -> mark#(cons(Y,first(X,Z)))
p6: mark#(dbl(X)) -> mark#(X)
p7: mark#(dbl(X)) -> active#(dbl(mark(X)))
p8: active#(add(s(X),Y)) -> mark#(s(add(X,Y)))
p9: mark#(add(X1,X2)) -> mark#(X2)
p10: mark#(add(X1,X2)) -> mark#(X1)
p11: mark#(add(X1,X2)) -> active#(add(mark(X1),mark(X2)))
p12: active#(add(|0|(),X)) -> mark#(X)
p13: mark#(s(X)) -> active#(s(X))
p14: active#(dbl(s(X))) -> mark#(s(s(dbl(X))))
p15: mark#(sqr(X)) -> mark#(X)
p16: mark#(sqr(X)) -> active#(sqr(mark(X)))
p17: active#(sqr(s(X))) -> mark#(s(add(sqr(X),dbl(X))))
p18: mark#(recip(X)) -> mark#(X)
p19: mark#(recip(X)) -> active#(recip(mark(X)))
p20: mark#(cons(X1,X2)) -> mark#(X1)
p21: mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2))
p22: mark#(terms(X)) -> mark#(X)
p23: mark#(terms(X)) -> active#(terms(mark(X)))

and R consists of:

r1: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
r2: active(sqr(|0|())) -> mark(|0|())
r3: active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
r4: active(dbl(|0|())) -> mark(|0|())
r5: active(dbl(s(X))) -> mark(s(s(dbl(X))))
r6: active(add(|0|(),X)) -> mark(X)
r7: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r8: active(first(|0|(),X)) -> mark(nil())
r9: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r10: mark(terms(X)) -> active(terms(mark(X)))
r11: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r12: mark(recip(X)) -> active(recip(mark(X)))
r13: mark(sqr(X)) -> active(sqr(mark(X)))
r14: mark(s(X)) -> active(s(X))
r15: mark(|0|()) -> active(|0|())
r16: mark(add(X1,X2)) -> active(add(mark(X1),mark(X2)))
r17: mark(dbl(X)) -> active(dbl(mark(X)))
r18: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r19: mark(nil()) -> active(nil())
r20: terms(mark(X)) -> terms(X)
r21: terms(active(X)) -> terms(X)
r22: cons(mark(X1),X2) -> cons(X1,X2)
r23: cons(X1,mark(X2)) -> cons(X1,X2)
r24: cons(active(X1),X2) -> cons(X1,X2)
r25: cons(X1,active(X2)) -> cons(X1,X2)
r26: recip(mark(X)) -> recip(X)
r27: recip(active(X)) -> recip(X)
r28: sqr(mark(X)) -> sqr(X)
r29: sqr(active(X)) -> sqr(X)
r30: s(mark(X)) -> s(X)
r31: s(active(X)) -> s(X)
r32: add(mark(X1),X2) -> add(X1,X2)
r33: add(X1,mark(X2)) -> add(X1,X2)
r34: add(active(X1),X2) -> add(X1,X2)
r35: add(X1,active(X2)) -> add(X1,X2)
r36: dbl(mark(X)) -> dbl(X)
r37: dbl(active(X)) -> dbl(X)
r38: first(mark(X1),X2) -> first(X1,X2)
r39: first(X1,mark(X2)) -> first(X1,X2)
r40: first(active(X1),X2) -> first(X1,X2)
r41: first(X1,active(X2)) -> first(X1,X2)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30, r31, r32, r33, r34, r35, r36, r37, r38, r39, r40, r41

Take the reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        active#_A(x1) = max{76, x1 - 40}
        terms_A(x1) = x1 + 387
        mark#_A(x1) = max{18, x1 + 14}
        cons_A(x1,x2) = x1 + 116
        recip_A(x1) = x1 + 76
        sqr_A(x1) = max{98, x1 + 59}
        s_A(x1) = 78
        first_A(x1,x2) = max{x1 + 101, x2 + 101}
        mark_A(x1) = x1
        dbl_A(x1) = x1 + 299
        add_A(x1,x2) = max{136, x1 + 133, x2 + 133}
        |0|_A = 99
        active_A(x1) = max{60, x1}
        nil_A = 100
  
    precedence:
    
      cons = sqr = mark = active = nil > |0| > active# = mark# = recip = s = first = dbl = add > terms
    
    partial status:
  
      pi(active#) = []
      pi(terms) = [1]
      pi(mark#) = [1]
      pi(cons) = []
      pi(recip) = [1]
      pi(sqr) = []
      pi(s) = []
      pi(first) = []
      pi(mark) = []
      pi(dbl) = []
      pi(add) = [1, 2]
      pi(|0|) = []
      pi(active) = []
      pi(nil) = []

The next rules are strictly ordered:

  p17

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: active#(terms(N)) -> mark#(cons(recip(sqr(N)),terms(s(N))))
p2: mark#(first(X1,X2)) -> mark#(X2)
p3: mark#(first(X1,X2)) -> mark#(X1)
p4: mark#(first(X1,X2)) -> active#(first(mark(X1),mark(X2)))
p5: active#(first(s(X),cons(Y,Z))) -> mark#(cons(Y,first(X,Z)))
p6: mark#(dbl(X)) -> mark#(X)
p7: mark#(dbl(X)) -> active#(dbl(mark(X)))
p8: active#(add(s(X),Y)) -> mark#(s(add(X,Y)))
p9: mark#(add(X1,X2)) -> mark#(X2)
p10: mark#(add(X1,X2)) -> mark#(X1)
p11: mark#(add(X1,X2)) -> active#(add(mark(X1),mark(X2)))
p12: active#(add(|0|(),X)) -> mark#(X)
p13: mark#(s(X)) -> active#(s(X))
p14: active#(dbl(s(X))) -> mark#(s(s(dbl(X))))
p15: mark#(sqr(X)) -> mark#(X)
p16: mark#(sqr(X)) -> active#(sqr(mark(X)))
p17: mark#(recip(X)) -> mark#(X)
p18: mark#(recip(X)) -> active#(recip(mark(X)))
p19: mark#(cons(X1,X2)) -> mark#(X1)
p20: mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2))
p21: mark#(terms(X)) -> mark#(X)
p22: mark#(terms(X)) -> active#(terms(mark(X)))

and R consists of:

r1: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
r2: active(sqr(|0|())) -> mark(|0|())
r3: active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
r4: active(dbl(|0|())) -> mark(|0|())
r5: active(dbl(s(X))) -> mark(s(s(dbl(X))))
r6: active(add(|0|(),X)) -> mark(X)
r7: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r8: active(first(|0|(),X)) -> mark(nil())
r9: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r10: mark(terms(X)) -> active(terms(mark(X)))
r11: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r12: mark(recip(X)) -> active(recip(mark(X)))
r13: mark(sqr(X)) -> active(sqr(mark(X)))
r14: mark(s(X)) -> active(s(X))
r15: mark(|0|()) -> active(|0|())
r16: mark(add(X1,X2)) -> active(add(mark(X1),mark(X2)))
r17: mark(dbl(X)) -> active(dbl(mark(X)))
r18: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r19: mark(nil()) -> active(nil())
r20: terms(mark(X)) -> terms(X)
r21: terms(active(X)) -> terms(X)
r22: cons(mark(X1),X2) -> cons(X1,X2)
r23: cons(X1,mark(X2)) -> cons(X1,X2)
r24: cons(active(X1),X2) -> cons(X1,X2)
r25: cons(X1,active(X2)) -> cons(X1,X2)
r26: recip(mark(X)) -> recip(X)
r27: recip(active(X)) -> recip(X)
r28: sqr(mark(X)) -> sqr(X)
r29: sqr(active(X)) -> sqr(X)
r30: s(mark(X)) -> s(X)
r31: s(active(X)) -> s(X)
r32: add(mark(X1),X2) -> add(X1,X2)
r33: add(X1,mark(X2)) -> add(X1,X2)
r34: add(active(X1),X2) -> add(X1,X2)
r35: add(X1,active(X2)) -> add(X1,X2)
r36: dbl(mark(X)) -> dbl(X)
r37: dbl(active(X)) -> dbl(X)
r38: first(mark(X1),X2) -> first(X1,X2)
r39: first(X1,mark(X2)) -> first(X1,X2)
r40: first(active(X1),X2) -> first(X1,X2)
r41: first(X1,active(X2)) -> first(X1,X2)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, p14, p15, p16, p17, p18, p19, p20, p21, p22}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: active#(terms(N)) -> mark#(cons(recip(sqr(N)),terms(s(N))))
p2: mark#(terms(X)) -> active#(terms(mark(X)))
p3: active#(dbl(s(X))) -> mark#(s(s(dbl(X))))
p4: mark#(terms(X)) -> mark#(X)
p5: mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2))
p6: active#(add(|0|(),X)) -> mark#(X)
p7: mark#(cons(X1,X2)) -> mark#(X1)
p8: mark#(recip(X)) -> active#(recip(mark(X)))
p9: active#(add(s(X),Y)) -> mark#(s(add(X,Y)))
p10: mark#(recip(X)) -> mark#(X)
p11: mark#(sqr(X)) -> active#(sqr(mark(X)))
p12: active#(first(s(X),cons(Y,Z))) -> mark#(cons(Y,first(X,Z)))
p13: mark#(sqr(X)) -> mark#(X)
p14: mark#(s(X)) -> active#(s(X))
p15: mark#(add(X1,X2)) -> active#(add(mark(X1),mark(X2)))
p16: mark#(add(X1,X2)) -> mark#(X1)
p17: mark#(add(X1,X2)) -> mark#(X2)
p18: mark#(dbl(X)) -> active#(dbl(mark(X)))
p19: mark#(dbl(X)) -> mark#(X)
p20: mark#(first(X1,X2)) -> active#(first(mark(X1),mark(X2)))
p21: mark#(first(X1,X2)) -> mark#(X1)
p22: mark#(first(X1,X2)) -> mark#(X2)

and R consists of:

r1: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
r2: active(sqr(|0|())) -> mark(|0|())
r3: active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
r4: active(dbl(|0|())) -> mark(|0|())
r5: active(dbl(s(X))) -> mark(s(s(dbl(X))))
r6: active(add(|0|(),X)) -> mark(X)
r7: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r8: active(first(|0|(),X)) -> mark(nil())
r9: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r10: mark(terms(X)) -> active(terms(mark(X)))
r11: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r12: mark(recip(X)) -> active(recip(mark(X)))
r13: mark(sqr(X)) -> active(sqr(mark(X)))
r14: mark(s(X)) -> active(s(X))
r15: mark(|0|()) -> active(|0|())
r16: mark(add(X1,X2)) -> active(add(mark(X1),mark(X2)))
r17: mark(dbl(X)) -> active(dbl(mark(X)))
r18: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r19: mark(nil()) -> active(nil())
r20: terms(mark(X)) -> terms(X)
r21: terms(active(X)) -> terms(X)
r22: cons(mark(X1),X2) -> cons(X1,X2)
r23: cons(X1,mark(X2)) -> cons(X1,X2)
r24: cons(active(X1),X2) -> cons(X1,X2)
r25: cons(X1,active(X2)) -> cons(X1,X2)
r26: recip(mark(X)) -> recip(X)
r27: recip(active(X)) -> recip(X)
r28: sqr(mark(X)) -> sqr(X)
r29: sqr(active(X)) -> sqr(X)
r30: s(mark(X)) -> s(X)
r31: s(active(X)) -> s(X)
r32: add(mark(X1),X2) -> add(X1,X2)
r33: add(X1,mark(X2)) -> add(X1,X2)
r34: add(active(X1),X2) -> add(X1,X2)
r35: add(X1,active(X2)) -> add(X1,X2)
r36: dbl(mark(X)) -> dbl(X)
r37: dbl(active(X)) -> dbl(X)
r38: first(mark(X1),X2) -> first(X1,X2)
r39: first(X1,mark(X2)) -> first(X1,X2)
r40: first(active(X1),X2) -> first(X1,X2)
r41: first(X1,active(X2)) -> first(X1,X2)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30, r31, r32, r33, r34, r35, r36, r37, r38, r39, r40, r41

Take the reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        active#_A(x1) = max{96, x1 - 127}
        terms_A(x1) = max{392, x1 + 366}
        mark#_A(x1) = x1 + 89
        cons_A(x1,x2) = max{176, x1}
        recip_A(x1) = max{45, x1}
        sqr_A(x1) = x1 + 21
        s_A(x1) = 7
        mark_A(x1) = x1 + 39
        dbl_A(x1) = x1 + 100
        add_A(x1,x2) = max{x1, x2 + 216}
        |0|_A = 7
        first_A(x1,x2) = max{x1 + 205, x2 + 216}
        active_A(x1) = max{46, x1}
        nil_A = 172
  
    precedence:
    
      terms = cons = recip = s = mark = active = nil > first > active# = mark# > sqr = |0| > dbl = add
    
    partial status:
  
      pi(active#) = []
      pi(terms) = []
      pi(mark#) = []
      pi(cons) = []
      pi(recip) = []
      pi(sqr) = []
      pi(s) = []
      pi(mark) = []
      pi(dbl) = [1]
      pi(add) = [2]
      pi(|0|) = []
      pi(first) = []
      pi(active) = []
      pi(nil) = []

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: active#(terms(N)) -> mark#(cons(recip(sqr(N)),terms(s(N))))
p2: active#(dbl(s(X))) -> mark#(s(s(dbl(X))))
p3: mark#(terms(X)) -> mark#(X)
p4: mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2))
p5: active#(add(|0|(),X)) -> mark#(X)
p6: mark#(cons(X1,X2)) -> mark#(X1)
p7: mark#(recip(X)) -> active#(recip(mark(X)))
p8: active#(add(s(X),Y)) -> mark#(s(add(X,Y)))
p9: mark#(recip(X)) -> mark#(X)
p10: mark#(sqr(X)) -> active#(sqr(mark(X)))
p11: active#(first(s(X),cons(Y,Z))) -> mark#(cons(Y,first(X,Z)))
p12: mark#(sqr(X)) -> mark#(X)
p13: mark#(s(X)) -> active#(s(X))
p14: mark#(add(X1,X2)) -> active#(add(mark(X1),mark(X2)))
p15: mark#(add(X1,X2)) -> mark#(X1)
p16: mark#(add(X1,X2)) -> mark#(X2)
p17: mark#(dbl(X)) -> active#(dbl(mark(X)))
p18: mark#(dbl(X)) -> mark#(X)
p19: mark#(first(X1,X2)) -> active#(first(mark(X1),mark(X2)))
p20: mark#(first(X1,X2)) -> mark#(X1)
p21: mark#(first(X1,X2)) -> mark#(X2)

and R consists of:

r1: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
r2: active(sqr(|0|())) -> mark(|0|())
r3: active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
r4: active(dbl(|0|())) -> mark(|0|())
r5: active(dbl(s(X))) -> mark(s(s(dbl(X))))
r6: active(add(|0|(),X)) -> mark(X)
r7: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r8: active(first(|0|(),X)) -> mark(nil())
r9: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r10: mark(terms(X)) -> active(terms(mark(X)))
r11: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r12: mark(recip(X)) -> active(recip(mark(X)))
r13: mark(sqr(X)) -> active(sqr(mark(X)))
r14: mark(s(X)) -> active(s(X))
r15: mark(|0|()) -> active(|0|())
r16: mark(add(X1,X2)) -> active(add(mark(X1),mark(X2)))
r17: mark(dbl(X)) -> active(dbl(mark(X)))
r18: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r19: mark(nil()) -> active(nil())
r20: terms(mark(X)) -> terms(X)
r21: terms(active(X)) -> terms(X)
r22: cons(mark(X1),X2) -> cons(X1,X2)
r23: cons(X1,mark(X2)) -> cons(X1,X2)
r24: cons(active(X1),X2) -> cons(X1,X2)
r25: cons(X1,active(X2)) -> cons(X1,X2)
r26: recip(mark(X)) -> recip(X)
r27: recip(active(X)) -> recip(X)
r28: sqr(mark(X)) -> sqr(X)
r29: sqr(active(X)) -> sqr(X)
r30: s(mark(X)) -> s(X)
r31: s(active(X)) -> s(X)
r32: add(mark(X1),X2) -> add(X1,X2)
r33: add(X1,mark(X2)) -> add(X1,X2)
r34: add(active(X1),X2) -> add(X1,X2)
r35: add(X1,active(X2)) -> add(X1,X2)
r36: dbl(mark(X)) -> dbl(X)
r37: dbl(active(X)) -> dbl(X)
r38: first(mark(X1),X2) -> first(X1,X2)
r39: first(X1,mark(X2)) -> first(X1,X2)
r40: first(active(X1),X2) -> first(X1,X2)
r41: first(X1,active(X2)) -> first(X1,X2)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, p14, p15, p16, p17, p18, p19, p20, p21}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: active#(terms(N)) -> mark#(cons(recip(sqr(N)),terms(s(N))))
p2: mark#(first(X1,X2)) -> mark#(X2)
p3: mark#(first(X1,X2)) -> mark#(X1)
p4: mark#(first(X1,X2)) -> active#(first(mark(X1),mark(X2)))
p5: active#(first(s(X),cons(Y,Z))) -> mark#(cons(Y,first(X,Z)))
p6: mark#(dbl(X)) -> mark#(X)
p7: mark#(dbl(X)) -> active#(dbl(mark(X)))
p8: active#(add(s(X),Y)) -> mark#(s(add(X,Y)))
p9: mark#(add(X1,X2)) -> mark#(X2)
p10: mark#(add(X1,X2)) -> mark#(X1)
p11: mark#(add(X1,X2)) -> active#(add(mark(X1),mark(X2)))
p12: active#(add(|0|(),X)) -> mark#(X)
p13: mark#(s(X)) -> active#(s(X))
p14: active#(dbl(s(X))) -> mark#(s(s(dbl(X))))
p15: mark#(sqr(X)) -> mark#(X)
p16: mark#(sqr(X)) -> active#(sqr(mark(X)))
p17: mark#(recip(X)) -> mark#(X)
p18: mark#(recip(X)) -> active#(recip(mark(X)))
p19: mark#(cons(X1,X2)) -> mark#(X1)
p20: mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2))
p21: mark#(terms(X)) -> mark#(X)

and R consists of:

r1: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
r2: active(sqr(|0|())) -> mark(|0|())
r3: active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
r4: active(dbl(|0|())) -> mark(|0|())
r5: active(dbl(s(X))) -> mark(s(s(dbl(X))))
r6: active(add(|0|(),X)) -> mark(X)
r7: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r8: active(first(|0|(),X)) -> mark(nil())
r9: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r10: mark(terms(X)) -> active(terms(mark(X)))
r11: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r12: mark(recip(X)) -> active(recip(mark(X)))
r13: mark(sqr(X)) -> active(sqr(mark(X)))
r14: mark(s(X)) -> active(s(X))
r15: mark(|0|()) -> active(|0|())
r16: mark(add(X1,X2)) -> active(add(mark(X1),mark(X2)))
r17: mark(dbl(X)) -> active(dbl(mark(X)))
r18: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r19: mark(nil()) -> active(nil())
r20: terms(mark(X)) -> terms(X)
r21: terms(active(X)) -> terms(X)
r22: cons(mark(X1),X2) -> cons(X1,X2)
r23: cons(X1,mark(X2)) -> cons(X1,X2)
r24: cons(active(X1),X2) -> cons(X1,X2)
r25: cons(X1,active(X2)) -> cons(X1,X2)
r26: recip(mark(X)) -> recip(X)
r27: recip(active(X)) -> recip(X)
r28: sqr(mark(X)) -> sqr(X)
r29: sqr(active(X)) -> sqr(X)
r30: s(mark(X)) -> s(X)
r31: s(active(X)) -> s(X)
r32: add(mark(X1),X2) -> add(X1,X2)
r33: add(X1,mark(X2)) -> add(X1,X2)
r34: add(active(X1),X2) -> add(X1,X2)
r35: add(X1,active(X2)) -> add(X1,X2)
r36: dbl(mark(X)) -> dbl(X)
r37: dbl(active(X)) -> dbl(X)
r38: first(mark(X1),X2) -> first(X1,X2)
r39: first(X1,mark(X2)) -> first(X1,X2)
r40: first(active(X1),X2) -> first(X1,X2)
r41: first(X1,active(X2)) -> first(X1,X2)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30, r31, r32, r33, r34, r35, r36, r37, r38, r39, r40, r41

Take the reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        active#_A(x1) = max{20, x1 - 22}
        terms_A(x1) = max{103, x1 + 63}
        mark#_A(x1) = x1 + 20
        cons_A(x1,x2) = x1 + 21
        recip_A(x1) = x1
        sqr_A(x1) = max{40, x1}
        s_A(x1) = 0
        first_A(x1,x2) = max{46, x1 + 38, x2 + 42}
        mark_A(x1) = x1 + 22
        dbl_A(x1) = x1
        add_A(x1,x2) = max{x1, x2 + 42}
        |0|_A = 0
        active_A(x1) = max{22, x1}
        nil_A = 23
  
    precedence:
    
      s = first > nil > active# = terms = mark# = cons = |0| > recip = sqr = mark = dbl = add = active
    
    partial status:
  
      pi(active#) = []
      pi(terms) = []
      pi(mark#) = []
      pi(cons) = []
      pi(recip) = []
      pi(sqr) = []
      pi(s) = []
      pi(first) = []
      pi(mark) = []
      pi(dbl) = []
      pi(add) = []
      pi(|0|) = []
      pi(active) = []
      pi(nil) = []

The next rules are strictly ordered:

  p3

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: active#(terms(N)) -> mark#(cons(recip(sqr(N)),terms(s(N))))
p2: mark#(first(X1,X2)) -> mark#(X2)
p3: mark#(first(X1,X2)) -> active#(first(mark(X1),mark(X2)))
p4: active#(first(s(X),cons(Y,Z))) -> mark#(cons(Y,first(X,Z)))
p5: mark#(dbl(X)) -> mark#(X)
p6: mark#(dbl(X)) -> active#(dbl(mark(X)))
p7: active#(add(s(X),Y)) -> mark#(s(add(X,Y)))
p8: mark#(add(X1,X2)) -> mark#(X2)
p9: mark#(add(X1,X2)) -> mark#(X1)
p10: mark#(add(X1,X2)) -> active#(add(mark(X1),mark(X2)))
p11: active#(add(|0|(),X)) -> mark#(X)
p12: mark#(s(X)) -> active#(s(X))
p13: active#(dbl(s(X))) -> mark#(s(s(dbl(X))))
p14: mark#(sqr(X)) -> mark#(X)
p15: mark#(sqr(X)) -> active#(sqr(mark(X)))
p16: mark#(recip(X)) -> mark#(X)
p17: mark#(recip(X)) -> active#(recip(mark(X)))
p18: mark#(cons(X1,X2)) -> mark#(X1)
p19: mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2))
p20: mark#(terms(X)) -> mark#(X)

and R consists of:

r1: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
r2: active(sqr(|0|())) -> mark(|0|())
r3: active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
r4: active(dbl(|0|())) -> mark(|0|())
r5: active(dbl(s(X))) -> mark(s(s(dbl(X))))
r6: active(add(|0|(),X)) -> mark(X)
r7: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r8: active(first(|0|(),X)) -> mark(nil())
r9: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r10: mark(terms(X)) -> active(terms(mark(X)))
r11: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r12: mark(recip(X)) -> active(recip(mark(X)))
r13: mark(sqr(X)) -> active(sqr(mark(X)))
r14: mark(s(X)) -> active(s(X))
r15: mark(|0|()) -> active(|0|())
r16: mark(add(X1,X2)) -> active(add(mark(X1),mark(X2)))
r17: mark(dbl(X)) -> active(dbl(mark(X)))
r18: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r19: mark(nil()) -> active(nil())
r20: terms(mark(X)) -> terms(X)
r21: terms(active(X)) -> terms(X)
r22: cons(mark(X1),X2) -> cons(X1,X2)
r23: cons(X1,mark(X2)) -> cons(X1,X2)
r24: cons(active(X1),X2) -> cons(X1,X2)
r25: cons(X1,active(X2)) -> cons(X1,X2)
r26: recip(mark(X)) -> recip(X)
r27: recip(active(X)) -> recip(X)
r28: sqr(mark(X)) -> sqr(X)
r29: sqr(active(X)) -> sqr(X)
r30: s(mark(X)) -> s(X)
r31: s(active(X)) -> s(X)
r32: add(mark(X1),X2) -> add(X1,X2)
r33: add(X1,mark(X2)) -> add(X1,X2)
r34: add(active(X1),X2) -> add(X1,X2)
r35: add(X1,active(X2)) -> add(X1,X2)
r36: dbl(mark(X)) -> dbl(X)
r37: dbl(active(X)) -> dbl(X)
r38: first(mark(X1),X2) -> first(X1,X2)
r39: first(X1,mark(X2)) -> first(X1,X2)
r40: first(active(X1),X2) -> first(X1,X2)
r41: first(X1,active(X2)) -> first(X1,X2)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, p14, p15, p16, p17, p18, p19, p20}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: active#(terms(N)) -> mark#(cons(recip(sqr(N)),terms(s(N))))
p2: mark#(terms(X)) -> mark#(X)
p3: mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2))
p4: active#(dbl(s(X))) -> mark#(s(s(dbl(X))))
p5: mark#(cons(X1,X2)) -> mark#(X1)
p6: mark#(recip(X)) -> active#(recip(mark(X)))
p7: active#(add(|0|(),X)) -> mark#(X)
p8: mark#(recip(X)) -> mark#(X)
p9: mark#(sqr(X)) -> active#(sqr(mark(X)))
p10: active#(add(s(X),Y)) -> mark#(s(add(X,Y)))
p11: mark#(sqr(X)) -> mark#(X)
p12: mark#(s(X)) -> active#(s(X))
p13: active#(first(s(X),cons(Y,Z))) -> mark#(cons(Y,first(X,Z)))
p14: mark#(add(X1,X2)) -> active#(add(mark(X1),mark(X2)))
p15: mark#(add(X1,X2)) -> mark#(X1)
p16: mark#(add(X1,X2)) -> mark#(X2)
p17: mark#(dbl(X)) -> active#(dbl(mark(X)))
p18: mark#(dbl(X)) -> mark#(X)
p19: mark#(first(X1,X2)) -> active#(first(mark(X1),mark(X2)))
p20: mark#(first(X1,X2)) -> mark#(X2)

and R consists of:

r1: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
r2: active(sqr(|0|())) -> mark(|0|())
r3: active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
r4: active(dbl(|0|())) -> mark(|0|())
r5: active(dbl(s(X))) -> mark(s(s(dbl(X))))
r6: active(add(|0|(),X)) -> mark(X)
r7: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r8: active(first(|0|(),X)) -> mark(nil())
r9: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r10: mark(terms(X)) -> active(terms(mark(X)))
r11: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r12: mark(recip(X)) -> active(recip(mark(X)))
r13: mark(sqr(X)) -> active(sqr(mark(X)))
r14: mark(s(X)) -> active(s(X))
r15: mark(|0|()) -> active(|0|())
r16: mark(add(X1,X2)) -> active(add(mark(X1),mark(X2)))
r17: mark(dbl(X)) -> active(dbl(mark(X)))
r18: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r19: mark(nil()) -> active(nil())
r20: terms(mark(X)) -> terms(X)
r21: terms(active(X)) -> terms(X)
r22: cons(mark(X1),X2) -> cons(X1,X2)
r23: cons(X1,mark(X2)) -> cons(X1,X2)
r24: cons(active(X1),X2) -> cons(X1,X2)
r25: cons(X1,active(X2)) -> cons(X1,X2)
r26: recip(mark(X)) -> recip(X)
r27: recip(active(X)) -> recip(X)
r28: sqr(mark(X)) -> sqr(X)
r29: sqr(active(X)) -> sqr(X)
r30: s(mark(X)) -> s(X)
r31: s(active(X)) -> s(X)
r32: add(mark(X1),X2) -> add(X1,X2)
r33: add(X1,mark(X2)) -> add(X1,X2)
r34: add(active(X1),X2) -> add(X1,X2)
r35: add(X1,active(X2)) -> add(X1,X2)
r36: dbl(mark(X)) -> dbl(X)
r37: dbl(active(X)) -> dbl(X)
r38: first(mark(X1),X2) -> first(X1,X2)
r39: first(X1,mark(X2)) -> first(X1,X2)
r40: first(active(X1),X2) -> first(X1,X2)
r41: first(X1,active(X2)) -> first(X1,X2)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30, r31, r32, r33, r34, r35, r36, r37, r38, r39, r40, r41

Take the reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        active#_A(x1) = max{14, x1 - 16}
        terms_A(x1) = x1 + 283
        mark#_A(x1) = max{14, x1 - 16}
        cons_A(x1,x2) = max{x1 + 205, x2 - 140}
        recip_A(x1) = x1 + 15
        sqr_A(x1) = x1 + 62
        s_A(x1) = 11
        mark_A(x1) = x1
        dbl_A(x1) = max{13, x1}
        add_A(x1,x2) = max{x1, x2 + 31}
        |0|_A = 12
        first_A(x1,x2) = max{550, x2 + 345}
        active_A(x1) = max{11, x1}
        nil_A = 344
  
    precedence:
    
      active# = terms = mark# = cons = recip = sqr = s = mark = dbl = add = |0| = first = active = nil
    
    partial status:
  
      pi(active#) = []
      pi(terms) = []
      pi(mark#) = []
      pi(cons) = []
      pi(recip) = []
      pi(sqr) = []
      pi(s) = []
      pi(mark) = [1]
      pi(dbl) = []
      pi(add) = []
      pi(|0|) = []
      pi(first) = []
      pi(active) = [1]
      pi(nil) = []

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(terms(X)) -> mark#(X)
p2: mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2))
p3: active#(dbl(s(X))) -> mark#(s(s(dbl(X))))
p4: mark#(cons(X1,X2)) -> mark#(X1)
p5: mark#(recip(X)) -> active#(recip(mark(X)))
p6: active#(add(|0|(),X)) -> mark#(X)
p7: mark#(recip(X)) -> mark#(X)
p8: mark#(sqr(X)) -> active#(sqr(mark(X)))
p9: active#(add(s(X),Y)) -> mark#(s(add(X,Y)))
p10: mark#(sqr(X)) -> mark#(X)
p11: mark#(s(X)) -> active#(s(X))
p12: active#(first(s(X),cons(Y,Z))) -> mark#(cons(Y,first(X,Z)))
p13: mark#(add(X1,X2)) -> active#(add(mark(X1),mark(X2)))
p14: mark#(add(X1,X2)) -> mark#(X1)
p15: mark#(add(X1,X2)) -> mark#(X2)
p16: mark#(dbl(X)) -> active#(dbl(mark(X)))
p17: mark#(dbl(X)) -> mark#(X)
p18: mark#(first(X1,X2)) -> active#(first(mark(X1),mark(X2)))
p19: mark#(first(X1,X2)) -> mark#(X2)

and R consists of:

r1: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
r2: active(sqr(|0|())) -> mark(|0|())
r3: active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
r4: active(dbl(|0|())) -> mark(|0|())
r5: active(dbl(s(X))) -> mark(s(s(dbl(X))))
r6: active(add(|0|(),X)) -> mark(X)
r7: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r8: active(first(|0|(),X)) -> mark(nil())
r9: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r10: mark(terms(X)) -> active(terms(mark(X)))
r11: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r12: mark(recip(X)) -> active(recip(mark(X)))
r13: mark(sqr(X)) -> active(sqr(mark(X)))
r14: mark(s(X)) -> active(s(X))
r15: mark(|0|()) -> active(|0|())
r16: mark(add(X1,X2)) -> active(add(mark(X1),mark(X2)))
r17: mark(dbl(X)) -> active(dbl(mark(X)))
r18: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r19: mark(nil()) -> active(nil())
r20: terms(mark(X)) -> terms(X)
r21: terms(active(X)) -> terms(X)
r22: cons(mark(X1),X2) -> cons(X1,X2)
r23: cons(X1,mark(X2)) -> cons(X1,X2)
r24: cons(active(X1),X2) -> cons(X1,X2)
r25: cons(X1,active(X2)) -> cons(X1,X2)
r26: recip(mark(X)) -> recip(X)
r27: recip(active(X)) -> recip(X)
r28: sqr(mark(X)) -> sqr(X)
r29: sqr(active(X)) -> sqr(X)
r30: s(mark(X)) -> s(X)
r31: s(active(X)) -> s(X)
r32: add(mark(X1),X2) -> add(X1,X2)
r33: add(X1,mark(X2)) -> add(X1,X2)
r34: add(active(X1),X2) -> add(X1,X2)
r35: add(X1,active(X2)) -> add(X1,X2)
r36: dbl(mark(X)) -> dbl(X)
r37: dbl(active(X)) -> dbl(X)
r38: first(mark(X1),X2) -> first(X1,X2)
r39: first(X1,mark(X2)) -> first(X1,X2)
r40: first(active(X1),X2) -> first(X1,X2)
r41: first(X1,active(X2)) -> first(X1,X2)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, p14, p15, p16, p17, p18, p19}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(terms(X)) -> mark#(X)
p2: mark#(first(X1,X2)) -> mark#(X2)
p3: mark#(first(X1,X2)) -> active#(first(mark(X1),mark(X2)))
p4: active#(first(s(X),cons(Y,Z))) -> mark#(cons(Y,first(X,Z)))
p5: mark#(dbl(X)) -> mark#(X)
p6: mark#(dbl(X)) -> active#(dbl(mark(X)))
p7: active#(add(s(X),Y)) -> mark#(s(add(X,Y)))
p8: mark#(add(X1,X2)) -> mark#(X2)
p9: mark#(add(X1,X2)) -> mark#(X1)
p10: mark#(add(X1,X2)) -> active#(add(mark(X1),mark(X2)))
p11: active#(add(|0|(),X)) -> mark#(X)
p12: mark#(s(X)) -> active#(s(X))
p13: active#(dbl(s(X))) -> mark#(s(s(dbl(X))))
p14: mark#(sqr(X)) -> mark#(X)
p15: mark#(sqr(X)) -> active#(sqr(mark(X)))
p16: mark#(recip(X)) -> mark#(X)
p17: mark#(recip(X)) -> active#(recip(mark(X)))
p18: mark#(cons(X1,X2)) -> mark#(X1)
p19: mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2))

and R consists of:

r1: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
r2: active(sqr(|0|())) -> mark(|0|())
r3: active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
r4: active(dbl(|0|())) -> mark(|0|())
r5: active(dbl(s(X))) -> mark(s(s(dbl(X))))
r6: active(add(|0|(),X)) -> mark(X)
r7: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r8: active(first(|0|(),X)) -> mark(nil())
r9: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r10: mark(terms(X)) -> active(terms(mark(X)))
r11: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r12: mark(recip(X)) -> active(recip(mark(X)))
r13: mark(sqr(X)) -> active(sqr(mark(X)))
r14: mark(s(X)) -> active(s(X))
r15: mark(|0|()) -> active(|0|())
r16: mark(add(X1,X2)) -> active(add(mark(X1),mark(X2)))
r17: mark(dbl(X)) -> active(dbl(mark(X)))
r18: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r19: mark(nil()) -> active(nil())
r20: terms(mark(X)) -> terms(X)
r21: terms(active(X)) -> terms(X)
r22: cons(mark(X1),X2) -> cons(X1,X2)
r23: cons(X1,mark(X2)) -> cons(X1,X2)
r24: cons(active(X1),X2) -> cons(X1,X2)
r25: cons(X1,active(X2)) -> cons(X1,X2)
r26: recip(mark(X)) -> recip(X)
r27: recip(active(X)) -> recip(X)
r28: sqr(mark(X)) -> sqr(X)
r29: sqr(active(X)) -> sqr(X)
r30: s(mark(X)) -> s(X)
r31: s(active(X)) -> s(X)
r32: add(mark(X1),X2) -> add(X1,X2)
r33: add(X1,mark(X2)) -> add(X1,X2)
r34: add(active(X1),X2) -> add(X1,X2)
r35: add(X1,active(X2)) -> add(X1,X2)
r36: dbl(mark(X)) -> dbl(X)
r37: dbl(active(X)) -> dbl(X)
r38: first(mark(X1),X2) -> first(X1,X2)
r39: first(X1,mark(X2)) -> first(X1,X2)
r40: first(active(X1),X2) -> first(X1,X2)
r41: first(X1,active(X2)) -> first(X1,X2)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30, r31, r32, r33, r34, r35, r36, r37, r38, r39, r40, r41

Take the reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        mark#_A(x1) = x1 + 7
        terms_A(x1) = x1 + 193
        first_A(x1,x2) = max{x1 + 251, x2 + 249}
        active#_A(x1) = max{6, x1 - 26}
        mark_A(x1) = x1 + 25
        s_A(x1) = 33
        cons_A(x1,x2) = x1 + 6
        dbl_A(x1) = x1 + 67
        add_A(x1,x2) = max{x1 + 1, x2 + 68}
        |0|_A = 31
        sqr_A(x1) = x1 + 161
        recip_A(x1) = max{6, x1}
        active_A(x1) = max{2, x1}
        nil_A = 242
  
    precedence:
    
      terms = mark > mark# = first = active# = s = cons = dbl = add = |0| = sqr = recip = active = nil
    
    partial status:
  
      pi(mark#) = [1]
      pi(terms) = [1]
      pi(first) = [2]
      pi(active#) = []
      pi(mark) = [1]
      pi(s) = []
      pi(cons) = []
      pi(dbl) = []
      pi(add) = []
      pi(|0|) = []
      pi(sqr) = []
      pi(recip) = [1]
      pi(active) = [1]
      pi(nil) = []

The next rules are strictly ordered:

  p19

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(terms(X)) -> mark#(X)
p2: mark#(first(X1,X2)) -> mark#(X2)
p3: mark#(first(X1,X2)) -> active#(first(mark(X1),mark(X2)))
p4: active#(first(s(X),cons(Y,Z))) -> mark#(cons(Y,first(X,Z)))
p5: mark#(dbl(X)) -> mark#(X)
p6: mark#(dbl(X)) -> active#(dbl(mark(X)))
p7: active#(add(s(X),Y)) -> mark#(s(add(X,Y)))
p8: mark#(add(X1,X2)) -> mark#(X2)
p9: mark#(add(X1,X2)) -> mark#(X1)
p10: mark#(add(X1,X2)) -> active#(add(mark(X1),mark(X2)))
p11: active#(add(|0|(),X)) -> mark#(X)
p12: mark#(s(X)) -> active#(s(X))
p13: active#(dbl(s(X))) -> mark#(s(s(dbl(X))))
p14: mark#(sqr(X)) -> mark#(X)
p15: mark#(sqr(X)) -> active#(sqr(mark(X)))
p16: mark#(recip(X)) -> mark#(X)
p17: mark#(recip(X)) -> active#(recip(mark(X)))
p18: mark#(cons(X1,X2)) -> mark#(X1)

and R consists of:

r1: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
r2: active(sqr(|0|())) -> mark(|0|())
r3: active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
r4: active(dbl(|0|())) -> mark(|0|())
r5: active(dbl(s(X))) -> mark(s(s(dbl(X))))
r6: active(add(|0|(),X)) -> mark(X)
r7: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r8: active(first(|0|(),X)) -> mark(nil())
r9: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r10: mark(terms(X)) -> active(terms(mark(X)))
r11: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r12: mark(recip(X)) -> active(recip(mark(X)))
r13: mark(sqr(X)) -> active(sqr(mark(X)))
r14: mark(s(X)) -> active(s(X))
r15: mark(|0|()) -> active(|0|())
r16: mark(add(X1,X2)) -> active(add(mark(X1),mark(X2)))
r17: mark(dbl(X)) -> active(dbl(mark(X)))
r18: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r19: mark(nil()) -> active(nil())
r20: terms(mark(X)) -> terms(X)
r21: terms(active(X)) -> terms(X)
r22: cons(mark(X1),X2) -> cons(X1,X2)
r23: cons(X1,mark(X2)) -> cons(X1,X2)
r24: cons(active(X1),X2) -> cons(X1,X2)
r25: cons(X1,active(X2)) -> cons(X1,X2)
r26: recip(mark(X)) -> recip(X)
r27: recip(active(X)) -> recip(X)
r28: sqr(mark(X)) -> sqr(X)
r29: sqr(active(X)) -> sqr(X)
r30: s(mark(X)) -> s(X)
r31: s(active(X)) -> s(X)
r32: add(mark(X1),X2) -> add(X1,X2)
r33: add(X1,mark(X2)) -> add(X1,X2)
r34: add(active(X1),X2) -> add(X1,X2)
r35: add(X1,active(X2)) -> add(X1,X2)
r36: dbl(mark(X)) -> dbl(X)
r37: dbl(active(X)) -> dbl(X)
r38: first(mark(X1),X2) -> first(X1,X2)
r39: first(X1,mark(X2)) -> first(X1,X2)
r40: first(active(X1),X2) -> first(X1,X2)
r41: first(X1,active(X2)) -> first(X1,X2)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, p14, p15, p16, p17, p18}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(terms(X)) -> mark#(X)
p2: mark#(cons(X1,X2)) -> mark#(X1)
p3: mark#(recip(X)) -> active#(recip(mark(X)))
p4: active#(dbl(s(X))) -> mark#(s(s(dbl(X))))
p5: mark#(recip(X)) -> mark#(X)
p6: mark#(sqr(X)) -> active#(sqr(mark(X)))
p7: active#(add(|0|(),X)) -> mark#(X)
p8: mark#(sqr(X)) -> mark#(X)
p9: mark#(s(X)) -> active#(s(X))
p10: active#(add(s(X),Y)) -> mark#(s(add(X,Y)))
p11: mark#(add(X1,X2)) -> active#(add(mark(X1),mark(X2)))
p12: active#(first(s(X),cons(Y,Z))) -> mark#(cons(Y,first(X,Z)))
p13: mark#(add(X1,X2)) -> mark#(X1)
p14: mark#(add(X1,X2)) -> mark#(X2)
p15: mark#(dbl(X)) -> active#(dbl(mark(X)))
p16: mark#(dbl(X)) -> mark#(X)
p17: mark#(first(X1,X2)) -> active#(first(mark(X1),mark(X2)))
p18: mark#(first(X1,X2)) -> mark#(X2)

and R consists of:

r1: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
r2: active(sqr(|0|())) -> mark(|0|())
r3: active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
r4: active(dbl(|0|())) -> mark(|0|())
r5: active(dbl(s(X))) -> mark(s(s(dbl(X))))
r6: active(add(|0|(),X)) -> mark(X)
r7: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r8: active(first(|0|(),X)) -> mark(nil())
r9: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r10: mark(terms(X)) -> active(terms(mark(X)))
r11: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r12: mark(recip(X)) -> active(recip(mark(X)))
r13: mark(sqr(X)) -> active(sqr(mark(X)))
r14: mark(s(X)) -> active(s(X))
r15: mark(|0|()) -> active(|0|())
r16: mark(add(X1,X2)) -> active(add(mark(X1),mark(X2)))
r17: mark(dbl(X)) -> active(dbl(mark(X)))
r18: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r19: mark(nil()) -> active(nil())
r20: terms(mark(X)) -> terms(X)
r21: terms(active(X)) -> terms(X)
r22: cons(mark(X1),X2) -> cons(X1,X2)
r23: cons(X1,mark(X2)) -> cons(X1,X2)
r24: cons(active(X1),X2) -> cons(X1,X2)
r25: cons(X1,active(X2)) -> cons(X1,X2)
r26: recip(mark(X)) -> recip(X)
r27: recip(active(X)) -> recip(X)
r28: sqr(mark(X)) -> sqr(X)
r29: sqr(active(X)) -> sqr(X)
r30: s(mark(X)) -> s(X)
r31: s(active(X)) -> s(X)
r32: add(mark(X1),X2) -> add(X1,X2)
r33: add(X1,mark(X2)) -> add(X1,X2)
r34: add(active(X1),X2) -> add(X1,X2)
r35: add(X1,active(X2)) -> add(X1,X2)
r36: dbl(mark(X)) -> dbl(X)
r37: dbl(active(X)) -> dbl(X)
r38: first(mark(X1),X2) -> first(X1,X2)
r39: first(X1,mark(X2)) -> first(X1,X2)
r40: first(active(X1),X2) -> first(X1,X2)
r41: first(X1,active(X2)) -> first(X1,X2)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30, r31, r32, r33, r34, r35, r36, r37, r38, r39, r40, r41

Take the reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        mark#_A(x1) = x1 + 16
        terms_A(x1) = x1 + 195
        cons_A(x1,x2) = x1 + 12
        recip_A(x1) = max{165, x1}
        active#_A(x1) = max{9, x1 - 2}
        mark_A(x1) = x1 + 18
        dbl_A(x1) = x1 + 101
        s_A(x1) = 0
        sqr_A(x1) = x1 + 165
        add_A(x1,x2) = max{26, x1 + 11, x2 + 18}
        |0|_A = 26
        first_A(x1,x2) = max{x1 + 26, x2 + 34}
        active_A(x1) = x1
        nil_A = 33
  
    precedence:
    
      recip > |0| > cons = s = add > mark# = terms = active# = mark = dbl = sqr = first = active = nil
    
    partial status:
  
      pi(mark#) = []
      pi(terms) = []
      pi(cons) = []
      pi(recip) = []
      pi(active#) = []
      pi(mark) = []
      pi(dbl) = []
      pi(s) = []
      pi(sqr) = []
      pi(add) = []
      pi(|0|) = []
      pi(first) = []
      pi(active) = []
      pi(nil) = []

The next rules are strictly ordered:

  p10

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(terms(X)) -> mark#(X)
p2: mark#(cons(X1,X2)) -> mark#(X1)
p3: mark#(recip(X)) -> active#(recip(mark(X)))
p4: active#(dbl(s(X))) -> mark#(s(s(dbl(X))))
p5: mark#(recip(X)) -> mark#(X)
p6: mark#(sqr(X)) -> active#(sqr(mark(X)))
p7: active#(add(|0|(),X)) -> mark#(X)
p8: mark#(sqr(X)) -> mark#(X)
p9: mark#(s(X)) -> active#(s(X))
p10: mark#(add(X1,X2)) -> active#(add(mark(X1),mark(X2)))
p11: active#(first(s(X),cons(Y,Z))) -> mark#(cons(Y,first(X,Z)))
p12: mark#(add(X1,X2)) -> mark#(X1)
p13: mark#(add(X1,X2)) -> mark#(X2)
p14: mark#(dbl(X)) -> active#(dbl(mark(X)))
p15: mark#(dbl(X)) -> mark#(X)
p16: mark#(first(X1,X2)) -> active#(first(mark(X1),mark(X2)))
p17: mark#(first(X1,X2)) -> mark#(X2)

and R consists of:

r1: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
r2: active(sqr(|0|())) -> mark(|0|())
r3: active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
r4: active(dbl(|0|())) -> mark(|0|())
r5: active(dbl(s(X))) -> mark(s(s(dbl(X))))
r6: active(add(|0|(),X)) -> mark(X)
r7: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r8: active(first(|0|(),X)) -> mark(nil())
r9: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r10: mark(terms(X)) -> active(terms(mark(X)))
r11: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r12: mark(recip(X)) -> active(recip(mark(X)))
r13: mark(sqr(X)) -> active(sqr(mark(X)))
r14: mark(s(X)) -> active(s(X))
r15: mark(|0|()) -> active(|0|())
r16: mark(add(X1,X2)) -> active(add(mark(X1),mark(X2)))
r17: mark(dbl(X)) -> active(dbl(mark(X)))
r18: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r19: mark(nil()) -> active(nil())
r20: terms(mark(X)) -> terms(X)
r21: terms(active(X)) -> terms(X)
r22: cons(mark(X1),X2) -> cons(X1,X2)
r23: cons(X1,mark(X2)) -> cons(X1,X2)
r24: cons(active(X1),X2) -> cons(X1,X2)
r25: cons(X1,active(X2)) -> cons(X1,X2)
r26: recip(mark(X)) -> recip(X)
r27: recip(active(X)) -> recip(X)
r28: sqr(mark(X)) -> sqr(X)
r29: sqr(active(X)) -> sqr(X)
r30: s(mark(X)) -> s(X)
r31: s(active(X)) -> s(X)
r32: add(mark(X1),X2) -> add(X1,X2)
r33: add(X1,mark(X2)) -> add(X1,X2)
r34: add(active(X1),X2) -> add(X1,X2)
r35: add(X1,active(X2)) -> add(X1,X2)
r36: dbl(mark(X)) -> dbl(X)
r37: dbl(active(X)) -> dbl(X)
r38: first(mark(X1),X2) -> first(X1,X2)
r39: first(X1,mark(X2)) -> first(X1,X2)
r40: first(active(X1),X2) -> first(X1,X2)
r41: first(X1,active(X2)) -> first(X1,X2)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, p14, p15, p16, p17}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(terms(X)) -> mark#(X)
p2: mark#(first(X1,X2)) -> mark#(X2)
p3: mark#(first(X1,X2)) -> active#(first(mark(X1),mark(X2)))
p4: active#(first(s(X),cons(Y,Z))) -> mark#(cons(Y,first(X,Z)))
p5: mark#(dbl(X)) -> mark#(X)
p6: mark#(dbl(X)) -> active#(dbl(mark(X)))
p7: active#(add(|0|(),X)) -> mark#(X)
p8: mark#(add(X1,X2)) -> mark#(X2)
p9: mark#(add(X1,X2)) -> mark#(X1)
p10: mark#(add(X1,X2)) -> active#(add(mark(X1),mark(X2)))
p11: active#(dbl(s(X))) -> mark#(s(s(dbl(X))))
p12: mark#(s(X)) -> active#(s(X))
p13: mark#(sqr(X)) -> mark#(X)
p14: mark#(sqr(X)) -> active#(sqr(mark(X)))
p15: mark#(recip(X)) -> mark#(X)
p16: mark#(recip(X)) -> active#(recip(mark(X)))
p17: mark#(cons(X1,X2)) -> mark#(X1)

and R consists of:

r1: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
r2: active(sqr(|0|())) -> mark(|0|())
r3: active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
r4: active(dbl(|0|())) -> mark(|0|())
r5: active(dbl(s(X))) -> mark(s(s(dbl(X))))
r6: active(add(|0|(),X)) -> mark(X)
r7: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r8: active(first(|0|(),X)) -> mark(nil())
r9: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r10: mark(terms(X)) -> active(terms(mark(X)))
r11: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r12: mark(recip(X)) -> active(recip(mark(X)))
r13: mark(sqr(X)) -> active(sqr(mark(X)))
r14: mark(s(X)) -> active(s(X))
r15: mark(|0|()) -> active(|0|())
r16: mark(add(X1,X2)) -> active(add(mark(X1),mark(X2)))
r17: mark(dbl(X)) -> active(dbl(mark(X)))
r18: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r19: mark(nil()) -> active(nil())
r20: terms(mark(X)) -> terms(X)
r21: terms(active(X)) -> terms(X)
r22: cons(mark(X1),X2) -> cons(X1,X2)
r23: cons(X1,mark(X2)) -> cons(X1,X2)
r24: cons(active(X1),X2) -> cons(X1,X2)
r25: cons(X1,active(X2)) -> cons(X1,X2)
r26: recip(mark(X)) -> recip(X)
r27: recip(active(X)) -> recip(X)
r28: sqr(mark(X)) -> sqr(X)
r29: sqr(active(X)) -> sqr(X)
r30: s(mark(X)) -> s(X)
r31: s(active(X)) -> s(X)
r32: add(mark(X1),X2) -> add(X1,X2)
r33: add(X1,mark(X2)) -> add(X1,X2)
r34: add(active(X1),X2) -> add(X1,X2)
r35: add(X1,active(X2)) -> add(X1,X2)
r36: dbl(mark(X)) -> dbl(X)
r37: dbl(active(X)) -> dbl(X)
r38: first(mark(X1),X2) -> first(X1,X2)
r39: first(X1,mark(X2)) -> first(X1,X2)
r40: first(active(X1),X2) -> first(X1,X2)
r41: first(X1,active(X2)) -> first(X1,X2)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30, r31, r32, r33, r34, r35, r36, r37, r38, r39, r40, r41

Take the reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        mark#_A(x1) = x1 + 5
        terms_A(x1) = max{78, x1 + 38}
        first_A(x1,x2) = max{46, x1 + 40, x2 + 38}
        active#_A(x1) = max{1, x1 - 4}
        mark_A(x1) = x1
        s_A(x1) = max{6, x1 - 9}
        cons_A(x1,x2) = x1 + 6
        dbl_A(x1) = x1 + 16
        add_A(x1,x2) = max{x1, x2 + 9}
        |0|_A = 11
        sqr_A(x1) = x1 + 25
        recip_A(x1) = max{71, x1 + 7}
        active_A(x1) = max{6, x1}
        nil_A = 37
  
    precedence:
    
      first = add = recip > active# = s = dbl > terms > mark# = mark = cons = |0| = sqr = active = nil
    
    partial status:
  
      pi(mark#) = [1]
      pi(terms) = []
      pi(first) = []
      pi(active#) = []
      pi(mark) = []
      pi(s) = []
      pi(cons) = []
      pi(dbl) = []
      pi(add) = []
      pi(|0|) = []
      pi(sqr) = []
      pi(recip) = []
      pi(active) = []
      pi(nil) = []

The next rules are strictly ordered:

  p16

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(terms(X)) -> mark#(X)
p2: mark#(first(X1,X2)) -> mark#(X2)
p3: mark#(first(X1,X2)) -> active#(first(mark(X1),mark(X2)))
p4: active#(first(s(X),cons(Y,Z))) -> mark#(cons(Y,first(X,Z)))
p5: mark#(dbl(X)) -> mark#(X)
p6: mark#(dbl(X)) -> active#(dbl(mark(X)))
p7: active#(add(|0|(),X)) -> mark#(X)
p8: mark#(add(X1,X2)) -> mark#(X2)
p9: mark#(add(X1,X2)) -> mark#(X1)
p10: mark#(add(X1,X2)) -> active#(add(mark(X1),mark(X2)))
p11: active#(dbl(s(X))) -> mark#(s(s(dbl(X))))
p12: mark#(s(X)) -> active#(s(X))
p13: mark#(sqr(X)) -> mark#(X)
p14: mark#(sqr(X)) -> active#(sqr(mark(X)))
p15: mark#(recip(X)) -> mark#(X)
p16: mark#(cons(X1,X2)) -> mark#(X1)

and R consists of:

r1: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
r2: active(sqr(|0|())) -> mark(|0|())
r3: active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
r4: active(dbl(|0|())) -> mark(|0|())
r5: active(dbl(s(X))) -> mark(s(s(dbl(X))))
r6: active(add(|0|(),X)) -> mark(X)
r7: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r8: active(first(|0|(),X)) -> mark(nil())
r9: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r10: mark(terms(X)) -> active(terms(mark(X)))
r11: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r12: mark(recip(X)) -> active(recip(mark(X)))
r13: mark(sqr(X)) -> active(sqr(mark(X)))
r14: mark(s(X)) -> active(s(X))
r15: mark(|0|()) -> active(|0|())
r16: mark(add(X1,X2)) -> active(add(mark(X1),mark(X2)))
r17: mark(dbl(X)) -> active(dbl(mark(X)))
r18: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r19: mark(nil()) -> active(nil())
r20: terms(mark(X)) -> terms(X)
r21: terms(active(X)) -> terms(X)
r22: cons(mark(X1),X2) -> cons(X1,X2)
r23: cons(X1,mark(X2)) -> cons(X1,X2)
r24: cons(active(X1),X2) -> cons(X1,X2)
r25: cons(X1,active(X2)) -> cons(X1,X2)
r26: recip(mark(X)) -> recip(X)
r27: recip(active(X)) -> recip(X)
r28: sqr(mark(X)) -> sqr(X)
r29: sqr(active(X)) -> sqr(X)
r30: s(mark(X)) -> s(X)
r31: s(active(X)) -> s(X)
r32: add(mark(X1),X2) -> add(X1,X2)
r33: add(X1,mark(X2)) -> add(X1,X2)
r34: add(active(X1),X2) -> add(X1,X2)
r35: add(X1,active(X2)) -> add(X1,X2)
r36: dbl(mark(X)) -> dbl(X)
r37: dbl(active(X)) -> dbl(X)
r38: first(mark(X1),X2) -> first(X1,X2)
r39: first(X1,mark(X2)) -> first(X1,X2)
r40: first(active(X1),X2) -> first(X1,X2)
r41: first(X1,active(X2)) -> first(X1,X2)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, p14, p15, p16}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(terms(X)) -> mark#(X)
p2: mark#(cons(X1,X2)) -> mark#(X1)
p3: mark#(recip(X)) -> mark#(X)
p4: mark#(sqr(X)) -> active#(sqr(mark(X)))
p5: active#(dbl(s(X))) -> mark#(s(s(dbl(X))))
p6: mark#(sqr(X)) -> mark#(X)
p7: mark#(s(X)) -> active#(s(X))
p8: active#(add(|0|(),X)) -> mark#(X)
p9: mark#(add(X1,X2)) -> active#(add(mark(X1),mark(X2)))
p10: active#(first(s(X),cons(Y,Z))) -> mark#(cons(Y,first(X,Z)))
p11: mark#(add(X1,X2)) -> mark#(X1)
p12: mark#(add(X1,X2)) -> mark#(X2)
p13: mark#(dbl(X)) -> active#(dbl(mark(X)))
p14: mark#(dbl(X)) -> mark#(X)
p15: mark#(first(X1,X2)) -> active#(first(mark(X1),mark(X2)))
p16: mark#(first(X1,X2)) -> mark#(X2)

and R consists of:

r1: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
r2: active(sqr(|0|())) -> mark(|0|())
r3: active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
r4: active(dbl(|0|())) -> mark(|0|())
r5: active(dbl(s(X))) -> mark(s(s(dbl(X))))
r6: active(add(|0|(),X)) -> mark(X)
r7: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r8: active(first(|0|(),X)) -> mark(nil())
r9: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r10: mark(terms(X)) -> active(terms(mark(X)))
r11: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r12: mark(recip(X)) -> active(recip(mark(X)))
r13: mark(sqr(X)) -> active(sqr(mark(X)))
r14: mark(s(X)) -> active(s(X))
r15: mark(|0|()) -> active(|0|())
r16: mark(add(X1,X2)) -> active(add(mark(X1),mark(X2)))
r17: mark(dbl(X)) -> active(dbl(mark(X)))
r18: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r19: mark(nil()) -> active(nil())
r20: terms(mark(X)) -> terms(X)
r21: terms(active(X)) -> terms(X)
r22: cons(mark(X1),X2) -> cons(X1,X2)
r23: cons(X1,mark(X2)) -> cons(X1,X2)
r24: cons(active(X1),X2) -> cons(X1,X2)
r25: cons(X1,active(X2)) -> cons(X1,X2)
r26: recip(mark(X)) -> recip(X)
r27: recip(active(X)) -> recip(X)
r28: sqr(mark(X)) -> sqr(X)
r29: sqr(active(X)) -> sqr(X)
r30: s(mark(X)) -> s(X)
r31: s(active(X)) -> s(X)
r32: add(mark(X1),X2) -> add(X1,X2)
r33: add(X1,mark(X2)) -> add(X1,X2)
r34: add(active(X1),X2) -> add(X1,X2)
r35: add(X1,active(X2)) -> add(X1,X2)
r36: dbl(mark(X)) -> dbl(X)
r37: dbl(active(X)) -> dbl(X)
r38: first(mark(X1),X2) -> first(X1,X2)
r39: first(X1,mark(X2)) -> first(X1,X2)
r40: first(active(X1),X2) -> first(X1,X2)
r41: first(X1,active(X2)) -> first(X1,X2)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30, r31, r32, r33, r34, r35, r36, r37, r38, r39, r40, r41

Take the reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        mark#_A(x1) = x1 + 65
        terms_A(x1) = max{86, x1 + 42}
        cons_A(x1,x2) = max{46, x1}
        recip_A(x1) = max{45, x1}
        sqr_A(x1) = max{46, x1 + 14}
        active#_A(x1) = max{75, x1 + 6}
        mark_A(x1) = x1 + 28
        dbl_A(x1) = x1 + 138
        s_A(x1) = 45
        add_A(x1,x2) = max{x1 + 12, x2 + 65}
        |0|_A = 45
        first_A(x1,x2) = max{x1 + 98, x2 + 155}
        active_A(x1) = max{73, x1}
        nil_A = 46
  
    precedence:
    
      mark# = recip = sqr = mark = dbl = |0| = active = nil > first > terms > cons = active# = s = add
    
    partial status:
  
      pi(mark#) = []
      pi(terms) = []
      pi(cons) = []
      pi(recip) = []
      pi(sqr) = []
      pi(active#) = [1]
      pi(mark) = []
      pi(dbl) = []
      pi(s) = []
      pi(add) = [1]
      pi(|0|) = []
      pi(first) = []
      pi(active) = []
      pi(nil) = []

The next rules are strictly ordered:

  p15

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(terms(X)) -> mark#(X)
p2: mark#(cons(X1,X2)) -> mark#(X1)
p3: mark#(recip(X)) -> mark#(X)
p4: mark#(sqr(X)) -> active#(sqr(mark(X)))
p5: active#(dbl(s(X))) -> mark#(s(s(dbl(X))))
p6: mark#(sqr(X)) -> mark#(X)
p7: mark#(s(X)) -> active#(s(X))
p8: active#(add(|0|(),X)) -> mark#(X)
p9: mark#(add(X1,X2)) -> active#(add(mark(X1),mark(X2)))
p10: active#(first(s(X),cons(Y,Z))) -> mark#(cons(Y,first(X,Z)))
p11: mark#(add(X1,X2)) -> mark#(X1)
p12: mark#(add(X1,X2)) -> mark#(X2)
p13: mark#(dbl(X)) -> active#(dbl(mark(X)))
p14: mark#(dbl(X)) -> mark#(X)
p15: mark#(first(X1,X2)) -> mark#(X2)

and R consists of:

r1: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
r2: active(sqr(|0|())) -> mark(|0|())
r3: active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
r4: active(dbl(|0|())) -> mark(|0|())
r5: active(dbl(s(X))) -> mark(s(s(dbl(X))))
r6: active(add(|0|(),X)) -> mark(X)
r7: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r8: active(first(|0|(),X)) -> mark(nil())
r9: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r10: mark(terms(X)) -> active(terms(mark(X)))
r11: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r12: mark(recip(X)) -> active(recip(mark(X)))
r13: mark(sqr(X)) -> active(sqr(mark(X)))
r14: mark(s(X)) -> active(s(X))
r15: mark(|0|()) -> active(|0|())
r16: mark(add(X1,X2)) -> active(add(mark(X1),mark(X2)))
r17: mark(dbl(X)) -> active(dbl(mark(X)))
r18: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r19: mark(nil()) -> active(nil())
r20: terms(mark(X)) -> terms(X)
r21: terms(active(X)) -> terms(X)
r22: cons(mark(X1),X2) -> cons(X1,X2)
r23: cons(X1,mark(X2)) -> cons(X1,X2)
r24: cons(active(X1),X2) -> cons(X1,X2)
r25: cons(X1,active(X2)) -> cons(X1,X2)
r26: recip(mark(X)) -> recip(X)
r27: recip(active(X)) -> recip(X)
r28: sqr(mark(X)) -> sqr(X)
r29: sqr(active(X)) -> sqr(X)
r30: s(mark(X)) -> s(X)
r31: s(active(X)) -> s(X)
r32: add(mark(X1),X2) -> add(X1,X2)
r33: add(X1,mark(X2)) -> add(X1,X2)
r34: add(active(X1),X2) -> add(X1,X2)
r35: add(X1,active(X2)) -> add(X1,X2)
r36: dbl(mark(X)) -> dbl(X)
r37: dbl(active(X)) -> dbl(X)
r38: first(mark(X1),X2) -> first(X1,X2)
r39: first(X1,mark(X2)) -> first(X1,X2)
r40: first(active(X1),X2) -> first(X1,X2)
r41: first(X1,active(X2)) -> first(X1,X2)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, p14, p15}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(terms(X)) -> mark#(X)
p2: mark#(first(X1,X2)) -> mark#(X2)
p3: mark#(dbl(X)) -> mark#(X)
p4: mark#(dbl(X)) -> active#(dbl(mark(X)))
p5: active#(first(s(X),cons(Y,Z))) -> mark#(cons(Y,first(X,Z)))
p6: mark#(add(X1,X2)) -> mark#(X2)
p7: mark#(add(X1,X2)) -> mark#(X1)
p8: mark#(add(X1,X2)) -> active#(add(mark(X1),mark(X2)))
p9: active#(add(|0|(),X)) -> mark#(X)
p10: mark#(s(X)) -> active#(s(X))
p11: active#(dbl(s(X))) -> mark#(s(s(dbl(X))))
p12: mark#(sqr(X)) -> mark#(X)
p13: mark#(sqr(X)) -> active#(sqr(mark(X)))
p14: mark#(recip(X)) -> mark#(X)
p15: mark#(cons(X1,X2)) -> mark#(X1)

and R consists of:

r1: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
r2: active(sqr(|0|())) -> mark(|0|())
r3: active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
r4: active(dbl(|0|())) -> mark(|0|())
r5: active(dbl(s(X))) -> mark(s(s(dbl(X))))
r6: active(add(|0|(),X)) -> mark(X)
r7: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r8: active(first(|0|(),X)) -> mark(nil())
r9: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r10: mark(terms(X)) -> active(terms(mark(X)))
r11: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r12: mark(recip(X)) -> active(recip(mark(X)))
r13: mark(sqr(X)) -> active(sqr(mark(X)))
r14: mark(s(X)) -> active(s(X))
r15: mark(|0|()) -> active(|0|())
r16: mark(add(X1,X2)) -> active(add(mark(X1),mark(X2)))
r17: mark(dbl(X)) -> active(dbl(mark(X)))
r18: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r19: mark(nil()) -> active(nil())
r20: terms(mark(X)) -> terms(X)
r21: terms(active(X)) -> terms(X)
r22: cons(mark(X1),X2) -> cons(X1,X2)
r23: cons(X1,mark(X2)) -> cons(X1,X2)
r24: cons(active(X1),X2) -> cons(X1,X2)
r25: cons(X1,active(X2)) -> cons(X1,X2)
r26: recip(mark(X)) -> recip(X)
r27: recip(active(X)) -> recip(X)
r28: sqr(mark(X)) -> sqr(X)
r29: sqr(active(X)) -> sqr(X)
r30: s(mark(X)) -> s(X)
r31: s(active(X)) -> s(X)
r32: add(mark(X1),X2) -> add(X1,X2)
r33: add(X1,mark(X2)) -> add(X1,X2)
r34: add(active(X1),X2) -> add(X1,X2)
r35: add(X1,active(X2)) -> add(X1,X2)
r36: dbl(mark(X)) -> dbl(X)
r37: dbl(active(X)) -> dbl(X)
r38: first(mark(X1),X2) -> first(X1,X2)
r39: first(X1,mark(X2)) -> first(X1,X2)
r40: first(active(X1),X2) -> first(X1,X2)
r41: first(X1,active(X2)) -> first(X1,X2)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30, r31, r32, r33, r34, r35, r36, r37, r38, r39, r40, r41

Take the reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        mark#_A(x1) = max{48, x1 + 14}
        terms_A(x1) = x1 + 348
        first_A(x1,x2) = max{x1 + 425, x2 + 428}
        dbl_A(x1) = x1 + 114
        active#_A(x1) = max{49, x1 + 7}
        mark_A(x1) = x1 + 6
        s_A(x1) = 50
        cons_A(x1,x2) = max{25, x1 + 18}
        add_A(x1,x2) = max{x1 + 107, x2 + 98}
        |0|_A = 11
        sqr_A(x1) = x1 + 321
        recip_A(x1) = max{47, x1 + 3}
        active_A(x1) = max{17, x1}
        nil_A = 400
  
    precedence:
    
      s > mark# = first = active# = sqr = recip > terms = mark = cons = add = active > dbl = nil > |0|
    
    partial status:
  
      pi(mark#) = []
      pi(terms) = []
      pi(first) = []
      pi(dbl) = []
      pi(active#) = [1]
      pi(mark) = []
      pi(s) = []
      pi(cons) = []
      pi(add) = []
      pi(|0|) = []
      pi(sqr) = []
      pi(recip) = []
      pi(active) = []
      pi(nil) = []

The next rules are strictly ordered:

  p5

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(terms(X)) -> mark#(X)
p2: mark#(first(X1,X2)) -> mark#(X2)
p3: mark#(dbl(X)) -> mark#(X)
p4: mark#(dbl(X)) -> active#(dbl(mark(X)))
p5: mark#(add(X1,X2)) -> mark#(X2)
p6: mark#(add(X1,X2)) -> mark#(X1)
p7: mark#(add(X1,X2)) -> active#(add(mark(X1),mark(X2)))
p8: active#(add(|0|(),X)) -> mark#(X)
p9: mark#(s(X)) -> active#(s(X))
p10: active#(dbl(s(X))) -> mark#(s(s(dbl(X))))
p11: mark#(sqr(X)) -> mark#(X)
p12: mark#(sqr(X)) -> active#(sqr(mark(X)))
p13: mark#(recip(X)) -> mark#(X)
p14: mark#(cons(X1,X2)) -> mark#(X1)

and R consists of:

r1: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
r2: active(sqr(|0|())) -> mark(|0|())
r3: active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
r4: active(dbl(|0|())) -> mark(|0|())
r5: active(dbl(s(X))) -> mark(s(s(dbl(X))))
r6: active(add(|0|(),X)) -> mark(X)
r7: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r8: active(first(|0|(),X)) -> mark(nil())
r9: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r10: mark(terms(X)) -> active(terms(mark(X)))
r11: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r12: mark(recip(X)) -> active(recip(mark(X)))
r13: mark(sqr(X)) -> active(sqr(mark(X)))
r14: mark(s(X)) -> active(s(X))
r15: mark(|0|()) -> active(|0|())
r16: mark(add(X1,X2)) -> active(add(mark(X1),mark(X2)))
r17: mark(dbl(X)) -> active(dbl(mark(X)))
r18: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r19: mark(nil()) -> active(nil())
r20: terms(mark(X)) -> terms(X)
r21: terms(active(X)) -> terms(X)
r22: cons(mark(X1),X2) -> cons(X1,X2)
r23: cons(X1,mark(X2)) -> cons(X1,X2)
r24: cons(active(X1),X2) -> cons(X1,X2)
r25: cons(X1,active(X2)) -> cons(X1,X2)
r26: recip(mark(X)) -> recip(X)
r27: recip(active(X)) -> recip(X)
r28: sqr(mark(X)) -> sqr(X)
r29: sqr(active(X)) -> sqr(X)
r30: s(mark(X)) -> s(X)
r31: s(active(X)) -> s(X)
r32: add(mark(X1),X2) -> add(X1,X2)
r33: add(X1,mark(X2)) -> add(X1,X2)
r34: add(active(X1),X2) -> add(X1,X2)
r35: add(X1,active(X2)) -> add(X1,X2)
r36: dbl(mark(X)) -> dbl(X)
r37: dbl(active(X)) -> dbl(X)
r38: first(mark(X1),X2) -> first(X1,X2)
r39: first(X1,mark(X2)) -> first(X1,X2)
r40: first(active(X1),X2) -> first(X1,X2)
r41: first(X1,active(X2)) -> first(X1,X2)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, p14}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(terms(X)) -> mark#(X)
p2: mark#(cons(X1,X2)) -> mark#(X1)
p3: mark#(recip(X)) -> mark#(X)
p4: mark#(sqr(X)) -> active#(sqr(mark(X)))
p5: active#(dbl(s(X))) -> mark#(s(s(dbl(X))))
p6: mark#(sqr(X)) -> mark#(X)
p7: mark#(s(X)) -> active#(s(X))
p8: active#(add(|0|(),X)) -> mark#(X)
p9: mark#(add(X1,X2)) -> active#(add(mark(X1),mark(X2)))
p10: mark#(add(X1,X2)) -> mark#(X1)
p11: mark#(add(X1,X2)) -> mark#(X2)
p12: mark#(dbl(X)) -> active#(dbl(mark(X)))
p13: mark#(dbl(X)) -> mark#(X)
p14: mark#(first(X1,X2)) -> mark#(X2)

and R consists of:

r1: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
r2: active(sqr(|0|())) -> mark(|0|())
r3: active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
r4: active(dbl(|0|())) -> mark(|0|())
r5: active(dbl(s(X))) -> mark(s(s(dbl(X))))
r6: active(add(|0|(),X)) -> mark(X)
r7: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r8: active(first(|0|(),X)) -> mark(nil())
r9: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r10: mark(terms(X)) -> active(terms(mark(X)))
r11: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r12: mark(recip(X)) -> active(recip(mark(X)))
r13: mark(sqr(X)) -> active(sqr(mark(X)))
r14: mark(s(X)) -> active(s(X))
r15: mark(|0|()) -> active(|0|())
r16: mark(add(X1,X2)) -> active(add(mark(X1),mark(X2)))
r17: mark(dbl(X)) -> active(dbl(mark(X)))
r18: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r19: mark(nil()) -> active(nil())
r20: terms(mark(X)) -> terms(X)
r21: terms(active(X)) -> terms(X)
r22: cons(mark(X1),X2) -> cons(X1,X2)
r23: cons(X1,mark(X2)) -> cons(X1,X2)
r24: cons(active(X1),X2) -> cons(X1,X2)
r25: cons(X1,active(X2)) -> cons(X1,X2)
r26: recip(mark(X)) -> recip(X)
r27: recip(active(X)) -> recip(X)
r28: sqr(mark(X)) -> sqr(X)
r29: sqr(active(X)) -> sqr(X)
r30: s(mark(X)) -> s(X)
r31: s(active(X)) -> s(X)
r32: add(mark(X1),X2) -> add(X1,X2)
r33: add(X1,mark(X2)) -> add(X1,X2)
r34: add(active(X1),X2) -> add(X1,X2)
r35: add(X1,active(X2)) -> add(X1,X2)
r36: dbl(mark(X)) -> dbl(X)
r37: dbl(active(X)) -> dbl(X)
r38: first(mark(X1),X2) -> first(X1,X2)
r39: first(X1,mark(X2)) -> first(X1,X2)
r40: first(active(X1),X2) -> first(X1,X2)
r41: first(X1,active(X2)) -> first(X1,X2)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30, r31, r32, r33, r34, r35, r36, r37, r38, r39, r40, r41

Take the reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        mark#_A(x1) = x1 + 39
        terms_A(x1) = x1 + 18
        cons_A(x1,x2) = max{8, x1}
        recip_A(x1) = x1 + 8
        sqr_A(x1) = x1 + 10
        active#_A(x1) = x1 + 39
        mark_A(x1) = x1
        dbl_A(x1) = x1 + 36
        s_A(x1) = 9
        add_A(x1,x2) = max{x1, x2 + 34}
        |0|_A = 23
        first_A(x1,x2) = max{x1 + 7, x2 + 25}
        active_A(x1) = max{8, x1}
        nil_A = 24
  
    precedence:
    
      mark# = terms = recip > cons = sqr = active# = mark = dbl = s = add = |0| = first = active = nil
    
    partial status:
  
      pi(mark#) = []
      pi(terms) = []
      pi(cons) = []
      pi(recip) = []
      pi(sqr) = []
      pi(active#) = [1]
      pi(mark) = [1]
      pi(dbl) = []
      pi(s) = []
      pi(add) = []
      pi(|0|) = []
      pi(first) = []
      pi(active) = [1]
      pi(nil) = []

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(cons(X1,X2)) -> mark#(X1)
p2: mark#(recip(X)) -> mark#(X)
p3: mark#(sqr(X)) -> active#(sqr(mark(X)))
p4: active#(dbl(s(X))) -> mark#(s(s(dbl(X))))
p5: mark#(sqr(X)) -> mark#(X)
p6: mark#(s(X)) -> active#(s(X))
p7: active#(add(|0|(),X)) -> mark#(X)
p8: mark#(add(X1,X2)) -> active#(add(mark(X1),mark(X2)))
p9: mark#(add(X1,X2)) -> mark#(X1)
p10: mark#(add(X1,X2)) -> mark#(X2)
p11: mark#(dbl(X)) -> active#(dbl(mark(X)))
p12: mark#(dbl(X)) -> mark#(X)
p13: mark#(first(X1,X2)) -> mark#(X2)

and R consists of:

r1: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
r2: active(sqr(|0|())) -> mark(|0|())
r3: active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
r4: active(dbl(|0|())) -> mark(|0|())
r5: active(dbl(s(X))) -> mark(s(s(dbl(X))))
r6: active(add(|0|(),X)) -> mark(X)
r7: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r8: active(first(|0|(),X)) -> mark(nil())
r9: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r10: mark(terms(X)) -> active(terms(mark(X)))
r11: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r12: mark(recip(X)) -> active(recip(mark(X)))
r13: mark(sqr(X)) -> active(sqr(mark(X)))
r14: mark(s(X)) -> active(s(X))
r15: mark(|0|()) -> active(|0|())
r16: mark(add(X1,X2)) -> active(add(mark(X1),mark(X2)))
r17: mark(dbl(X)) -> active(dbl(mark(X)))
r18: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r19: mark(nil()) -> active(nil())
r20: terms(mark(X)) -> terms(X)
r21: terms(active(X)) -> terms(X)
r22: cons(mark(X1),X2) -> cons(X1,X2)
r23: cons(X1,mark(X2)) -> cons(X1,X2)
r24: cons(active(X1),X2) -> cons(X1,X2)
r25: cons(X1,active(X2)) -> cons(X1,X2)
r26: recip(mark(X)) -> recip(X)
r27: recip(active(X)) -> recip(X)
r28: sqr(mark(X)) -> sqr(X)
r29: sqr(active(X)) -> sqr(X)
r30: s(mark(X)) -> s(X)
r31: s(active(X)) -> s(X)
r32: add(mark(X1),X2) -> add(X1,X2)
r33: add(X1,mark(X2)) -> add(X1,X2)
r34: add(active(X1),X2) -> add(X1,X2)
r35: add(X1,active(X2)) -> add(X1,X2)
r36: dbl(mark(X)) -> dbl(X)
r37: dbl(active(X)) -> dbl(X)
r38: first(mark(X1),X2) -> first(X1,X2)
r39: first(X1,mark(X2)) -> first(X1,X2)
r40: first(active(X1),X2) -> first(X1,X2)
r41: first(X1,active(X2)) -> first(X1,X2)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(cons(X1,X2)) -> mark#(X1)
p2: mark#(first(X1,X2)) -> mark#(X2)
p3: mark#(dbl(X)) -> mark#(X)
p4: mark#(dbl(X)) -> active#(dbl(mark(X)))
p5: active#(add(|0|(),X)) -> mark#(X)
p6: mark#(add(X1,X2)) -> mark#(X2)
p7: mark#(add(X1,X2)) -> mark#(X1)
p8: mark#(add(X1,X2)) -> active#(add(mark(X1),mark(X2)))
p9: active#(dbl(s(X))) -> mark#(s(s(dbl(X))))
p10: mark#(s(X)) -> active#(s(X))
p11: mark#(sqr(X)) -> mark#(X)
p12: mark#(sqr(X)) -> active#(sqr(mark(X)))
p13: mark#(recip(X)) -> mark#(X)

and R consists of:

r1: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
r2: active(sqr(|0|())) -> mark(|0|())
r3: active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
r4: active(dbl(|0|())) -> mark(|0|())
r5: active(dbl(s(X))) -> mark(s(s(dbl(X))))
r6: active(add(|0|(),X)) -> mark(X)
r7: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r8: active(first(|0|(),X)) -> mark(nil())
r9: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r10: mark(terms(X)) -> active(terms(mark(X)))
r11: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r12: mark(recip(X)) -> active(recip(mark(X)))
r13: mark(sqr(X)) -> active(sqr(mark(X)))
r14: mark(s(X)) -> active(s(X))
r15: mark(|0|()) -> active(|0|())
r16: mark(add(X1,X2)) -> active(add(mark(X1),mark(X2)))
r17: mark(dbl(X)) -> active(dbl(mark(X)))
r18: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r19: mark(nil()) -> active(nil())
r20: terms(mark(X)) -> terms(X)
r21: terms(active(X)) -> terms(X)
r22: cons(mark(X1),X2) -> cons(X1,X2)
r23: cons(X1,mark(X2)) -> cons(X1,X2)
r24: cons(active(X1),X2) -> cons(X1,X2)
r25: cons(X1,active(X2)) -> cons(X1,X2)
r26: recip(mark(X)) -> recip(X)
r27: recip(active(X)) -> recip(X)
r28: sqr(mark(X)) -> sqr(X)
r29: sqr(active(X)) -> sqr(X)
r30: s(mark(X)) -> s(X)
r31: s(active(X)) -> s(X)
r32: add(mark(X1),X2) -> add(X1,X2)
r33: add(X1,mark(X2)) -> add(X1,X2)
r34: add(active(X1),X2) -> add(X1,X2)
r35: add(X1,active(X2)) -> add(X1,X2)
r36: dbl(mark(X)) -> dbl(X)
r37: dbl(active(X)) -> dbl(X)
r38: first(mark(X1),X2) -> first(X1,X2)
r39: first(X1,mark(X2)) -> first(X1,X2)
r40: first(active(X1),X2) -> first(X1,X2)
r41: first(X1,active(X2)) -> first(X1,X2)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30, r31, r32, r33, r34, r35, r36, r37, r38, r39, r40, r41

Take the reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        mark#_A(x1) = x1 + 19
        cons_A(x1,x2) = max{x1 + 92, x2 - 54}
        first_A(x1,x2) = max{111, x2 + 19}
        dbl_A(x1) = max{32, x1 + 19}
        active#_A(x1) = x1 + 12
        mark_A(x1) = x1
        add_A(x1,x2) = max{25, x1, x2 + 19}
        |0|_A = 26
        s_A(x1) = max{53, x1 - 29}
        sqr_A(x1) = x1 + 81
        recip_A(x1) = x1 + 27
        active_A(x1) = max{25, x1}
        terms_A(x1) = x1 + 200
        nil_A = 27
  
    precedence:
    
      add = |0| > first > dbl = s > recip > terms > mark# = active# = nil > cons = mark = sqr = active
    
    partial status:
  
      pi(mark#) = [1]
      pi(cons) = []
      pi(first) = []
      pi(dbl) = []
      pi(active#) = [1]
      pi(mark) = []
      pi(add) = []
      pi(|0|) = []
      pi(s) = []
      pi(sqr) = []
      pi(recip) = []
      pi(active) = []
      pi(terms) = []
      pi(nil) = []

The next rules are strictly ordered:

  p5

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(cons(X1,X2)) -> mark#(X1)
p2: mark#(first(X1,X2)) -> mark#(X2)
p3: mark#(dbl(X)) -> mark#(X)
p4: mark#(dbl(X)) -> active#(dbl(mark(X)))
p5: mark#(add(X1,X2)) -> mark#(X2)
p6: mark#(add(X1,X2)) -> mark#(X1)
p7: mark#(add(X1,X2)) -> active#(add(mark(X1),mark(X2)))
p8: active#(dbl(s(X))) -> mark#(s(s(dbl(X))))
p9: mark#(s(X)) -> active#(s(X))
p10: mark#(sqr(X)) -> mark#(X)
p11: mark#(sqr(X)) -> active#(sqr(mark(X)))
p12: mark#(recip(X)) -> mark#(X)

and R consists of:

r1: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
r2: active(sqr(|0|())) -> mark(|0|())
r3: active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
r4: active(dbl(|0|())) -> mark(|0|())
r5: active(dbl(s(X))) -> mark(s(s(dbl(X))))
r6: active(add(|0|(),X)) -> mark(X)
r7: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r8: active(first(|0|(),X)) -> mark(nil())
r9: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r10: mark(terms(X)) -> active(terms(mark(X)))
r11: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r12: mark(recip(X)) -> active(recip(mark(X)))
r13: mark(sqr(X)) -> active(sqr(mark(X)))
r14: mark(s(X)) -> active(s(X))
r15: mark(|0|()) -> active(|0|())
r16: mark(add(X1,X2)) -> active(add(mark(X1),mark(X2)))
r17: mark(dbl(X)) -> active(dbl(mark(X)))
r18: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r19: mark(nil()) -> active(nil())
r20: terms(mark(X)) -> terms(X)
r21: terms(active(X)) -> terms(X)
r22: cons(mark(X1),X2) -> cons(X1,X2)
r23: cons(X1,mark(X2)) -> cons(X1,X2)
r24: cons(active(X1),X2) -> cons(X1,X2)
r25: cons(X1,active(X2)) -> cons(X1,X2)
r26: recip(mark(X)) -> recip(X)
r27: recip(active(X)) -> recip(X)
r28: sqr(mark(X)) -> sqr(X)
r29: sqr(active(X)) -> sqr(X)
r30: s(mark(X)) -> s(X)
r31: s(active(X)) -> s(X)
r32: add(mark(X1),X2) -> add(X1,X2)
r33: add(X1,mark(X2)) -> add(X1,X2)
r34: add(active(X1),X2) -> add(X1,X2)
r35: add(X1,active(X2)) -> add(X1,X2)
r36: dbl(mark(X)) -> dbl(X)
r37: dbl(active(X)) -> dbl(X)
r38: first(mark(X1),X2) -> first(X1,X2)
r39: first(X1,mark(X2)) -> first(X1,X2)
r40: first(active(X1),X2) -> first(X1,X2)
r41: first(X1,active(X2)) -> first(X1,X2)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(cons(X1,X2)) -> mark#(X1)
p2: mark#(recip(X)) -> mark#(X)
p3: mark#(sqr(X)) -> active#(sqr(mark(X)))
p4: active#(dbl(s(X))) -> mark#(s(s(dbl(X))))
p5: mark#(sqr(X)) -> mark#(X)
p6: mark#(s(X)) -> active#(s(X))
p7: mark#(add(X1,X2)) -> active#(add(mark(X1),mark(X2)))
p8: mark#(add(X1,X2)) -> mark#(X1)
p9: mark#(add(X1,X2)) -> mark#(X2)
p10: mark#(dbl(X)) -> active#(dbl(mark(X)))
p11: mark#(dbl(X)) -> mark#(X)
p12: mark#(first(X1,X2)) -> mark#(X2)

and R consists of:

r1: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
r2: active(sqr(|0|())) -> mark(|0|())
r3: active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
r4: active(dbl(|0|())) -> mark(|0|())
r5: active(dbl(s(X))) -> mark(s(s(dbl(X))))
r6: active(add(|0|(),X)) -> mark(X)
r7: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r8: active(first(|0|(),X)) -> mark(nil())
r9: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r10: mark(terms(X)) -> active(terms(mark(X)))
r11: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r12: mark(recip(X)) -> active(recip(mark(X)))
r13: mark(sqr(X)) -> active(sqr(mark(X)))
r14: mark(s(X)) -> active(s(X))
r15: mark(|0|()) -> active(|0|())
r16: mark(add(X1,X2)) -> active(add(mark(X1),mark(X2)))
r17: mark(dbl(X)) -> active(dbl(mark(X)))
r18: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r19: mark(nil()) -> active(nil())
r20: terms(mark(X)) -> terms(X)
r21: terms(active(X)) -> terms(X)
r22: cons(mark(X1),X2) -> cons(X1,X2)
r23: cons(X1,mark(X2)) -> cons(X1,X2)
r24: cons(active(X1),X2) -> cons(X1,X2)
r25: cons(X1,active(X2)) -> cons(X1,X2)
r26: recip(mark(X)) -> recip(X)
r27: recip(active(X)) -> recip(X)
r28: sqr(mark(X)) -> sqr(X)
r29: sqr(active(X)) -> sqr(X)
r30: s(mark(X)) -> s(X)
r31: s(active(X)) -> s(X)
r32: add(mark(X1),X2) -> add(X1,X2)
r33: add(X1,mark(X2)) -> add(X1,X2)
r34: add(active(X1),X2) -> add(X1,X2)
r35: add(X1,active(X2)) -> add(X1,X2)
r36: dbl(mark(X)) -> dbl(X)
r37: dbl(active(X)) -> dbl(X)
r38: first(mark(X1),X2) -> first(X1,X2)
r39: first(X1,mark(X2)) -> first(X1,X2)
r40: first(active(X1),X2) -> first(X1,X2)
r41: first(X1,active(X2)) -> first(X1,X2)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30, r31, r32, r33, r34, r35, r36, r37, r38, r39, r40, r41

Take the reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        mark#_A(x1) = x1 + 8
        cons_A(x1,x2) = max{x1 + 9, x2}
        recip_A(x1) = x1 + 9
        sqr_A(x1) = max{57, x1 + 56}
        active#_A(x1) = max{5, x1 - 8}
        mark_A(x1) = x1
        dbl_A(x1) = x1 + 37
        s_A(x1) = 0
        add_A(x1,x2) = max{x1 + 13, x2 + 20}
        first_A(x1,x2) = x2 + 20
        active_A(x1) = x1
        terms_A(x1) = x1 + 77
        |0|_A = 18
        nil_A = 19
  
    precedence:
    
      mark = active = terms > mark# = recip = sqr = active# > cons > add = first = |0| = nil > dbl = s
    
    partial status:
  
      pi(mark#) = [1]
      pi(cons) = [1]
      pi(recip) = []
      pi(sqr) = []
      pi(active#) = []
      pi(mark) = []
      pi(dbl) = []
      pi(s) = []
      pi(add) = [1]
      pi(first) = [2]
      pi(active) = []
      pi(terms) = []
      pi(|0|) = []
      pi(nil) = []

The next rules are strictly ordered:

  p10

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(cons(X1,X2)) -> mark#(X1)
p2: mark#(recip(X)) -> mark#(X)
p3: mark#(sqr(X)) -> active#(sqr(mark(X)))
p4: active#(dbl(s(X))) -> mark#(s(s(dbl(X))))
p5: mark#(sqr(X)) -> mark#(X)
p6: mark#(s(X)) -> active#(s(X))
p7: mark#(add(X1,X2)) -> active#(add(mark(X1),mark(X2)))
p8: mark#(add(X1,X2)) -> mark#(X1)
p9: mark#(add(X1,X2)) -> mark#(X2)
p10: mark#(dbl(X)) -> mark#(X)
p11: mark#(first(X1,X2)) -> mark#(X2)

and R consists of:

r1: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
r2: active(sqr(|0|())) -> mark(|0|())
r3: active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
r4: active(dbl(|0|())) -> mark(|0|())
r5: active(dbl(s(X))) -> mark(s(s(dbl(X))))
r6: active(add(|0|(),X)) -> mark(X)
r7: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r8: active(first(|0|(),X)) -> mark(nil())
r9: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r10: mark(terms(X)) -> active(terms(mark(X)))
r11: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r12: mark(recip(X)) -> active(recip(mark(X)))
r13: mark(sqr(X)) -> active(sqr(mark(X)))
r14: mark(s(X)) -> active(s(X))
r15: mark(|0|()) -> active(|0|())
r16: mark(add(X1,X2)) -> active(add(mark(X1),mark(X2)))
r17: mark(dbl(X)) -> active(dbl(mark(X)))
r18: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r19: mark(nil()) -> active(nil())
r20: terms(mark(X)) -> terms(X)
r21: terms(active(X)) -> terms(X)
r22: cons(mark(X1),X2) -> cons(X1,X2)
r23: cons(X1,mark(X2)) -> cons(X1,X2)
r24: cons(active(X1),X2) -> cons(X1,X2)
r25: cons(X1,active(X2)) -> cons(X1,X2)
r26: recip(mark(X)) -> recip(X)
r27: recip(active(X)) -> recip(X)
r28: sqr(mark(X)) -> sqr(X)
r29: sqr(active(X)) -> sqr(X)
r30: s(mark(X)) -> s(X)
r31: s(active(X)) -> s(X)
r32: add(mark(X1),X2) -> add(X1,X2)
r33: add(X1,mark(X2)) -> add(X1,X2)
r34: add(active(X1),X2) -> add(X1,X2)
r35: add(X1,active(X2)) -> add(X1,X2)
r36: dbl(mark(X)) -> dbl(X)
r37: dbl(active(X)) -> dbl(X)
r38: first(mark(X1),X2) -> first(X1,X2)
r39: first(X1,mark(X2)) -> first(X1,X2)
r40: first(active(X1),X2) -> first(X1,X2)
r41: first(X1,active(X2)) -> first(X1,X2)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(cons(X1,X2)) -> mark#(X1)
p2: mark#(first(X1,X2)) -> mark#(X2)
p3: mark#(dbl(X)) -> mark#(X)
p4: mark#(add(X1,X2)) -> mark#(X2)
p5: mark#(add(X1,X2)) -> mark#(X1)
p6: mark#(add(X1,X2)) -> active#(add(mark(X1),mark(X2)))
p7: active#(dbl(s(X))) -> mark#(s(s(dbl(X))))
p8: mark#(s(X)) -> active#(s(X))
p9: mark#(sqr(X)) -> mark#(X)
p10: mark#(sqr(X)) -> active#(sqr(mark(X)))
p11: mark#(recip(X)) -> mark#(X)

and R consists of:

r1: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
r2: active(sqr(|0|())) -> mark(|0|())
r3: active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
r4: active(dbl(|0|())) -> mark(|0|())
r5: active(dbl(s(X))) -> mark(s(s(dbl(X))))
r6: active(add(|0|(),X)) -> mark(X)
r7: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r8: active(first(|0|(),X)) -> mark(nil())
r9: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r10: mark(terms(X)) -> active(terms(mark(X)))
r11: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r12: mark(recip(X)) -> active(recip(mark(X)))
r13: mark(sqr(X)) -> active(sqr(mark(X)))
r14: mark(s(X)) -> active(s(X))
r15: mark(|0|()) -> active(|0|())
r16: mark(add(X1,X2)) -> active(add(mark(X1),mark(X2)))
r17: mark(dbl(X)) -> active(dbl(mark(X)))
r18: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r19: mark(nil()) -> active(nil())
r20: terms(mark(X)) -> terms(X)
r21: terms(active(X)) -> terms(X)
r22: cons(mark(X1),X2) -> cons(X1,X2)
r23: cons(X1,mark(X2)) -> cons(X1,X2)
r24: cons(active(X1),X2) -> cons(X1,X2)
r25: cons(X1,active(X2)) -> cons(X1,X2)
r26: recip(mark(X)) -> recip(X)
r27: recip(active(X)) -> recip(X)
r28: sqr(mark(X)) -> sqr(X)
r29: sqr(active(X)) -> sqr(X)
r30: s(mark(X)) -> s(X)
r31: s(active(X)) -> s(X)
r32: add(mark(X1),X2) -> add(X1,X2)
r33: add(X1,mark(X2)) -> add(X1,X2)
r34: add(active(X1),X2) -> add(X1,X2)
r35: add(X1,active(X2)) -> add(X1,X2)
r36: dbl(mark(X)) -> dbl(X)
r37: dbl(active(X)) -> dbl(X)
r38: first(mark(X1),X2) -> first(X1,X2)
r39: first(X1,mark(X2)) -> first(X1,X2)
r40: first(active(X1),X2) -> first(X1,X2)
r41: first(X1,active(X2)) -> first(X1,X2)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30, r31, r32, r33, r34, r35, r36, r37, r38, r39, r40, r41

Take the reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        mark#_A(x1) = x1 + 18
        cons_A(x1,x2) = x1 + 5
        first_A(x1,x2) = max{x1 + 18, x2}
        dbl_A(x1) = x1 + 6
        add_A(x1,x2) = max{x1, x2 + 8}
        active#_A(x1) = x1 + 18
        mark_A(x1) = x1
        s_A(x1) = max{4, x1 - 1}
        sqr_A(x1) = x1 + 14
        recip_A(x1) = x1 + 4
        active_A(x1) = max{2, x1}
        terms_A(x1) = x1 + 24
        |0|_A = 2
        nil_A = 3
  
    precedence:
    
      dbl = mark = active = terms = |0| > mark# = cons = first = active# > s > add = sqr = recip = nil
    
    partial status:
  
      pi(mark#) = []
      pi(cons) = [1]
      pi(first) = [1]
      pi(dbl) = []
      pi(add) = [2]
      pi(active#) = []
      pi(mark) = []
      pi(s) = []
      pi(sqr) = [1]
      pi(recip) = [1]
      pi(active) = []
      pi(terms) = []
      pi(|0|) = []
      pi(nil) = []

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(first(X1,X2)) -> mark#(X2)
p2: mark#(dbl(X)) -> mark#(X)
p3: mark#(add(X1,X2)) -> mark#(X2)
p4: mark#(add(X1,X2)) -> mark#(X1)
p5: mark#(add(X1,X2)) -> active#(add(mark(X1),mark(X2)))
p6: active#(dbl(s(X))) -> mark#(s(s(dbl(X))))
p7: mark#(s(X)) -> active#(s(X))
p8: mark#(sqr(X)) -> mark#(X)
p9: mark#(sqr(X)) -> active#(sqr(mark(X)))
p10: mark#(recip(X)) -> mark#(X)

and R consists of:

r1: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
r2: active(sqr(|0|())) -> mark(|0|())
r3: active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
r4: active(dbl(|0|())) -> mark(|0|())
r5: active(dbl(s(X))) -> mark(s(s(dbl(X))))
r6: active(add(|0|(),X)) -> mark(X)
r7: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r8: active(first(|0|(),X)) -> mark(nil())
r9: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r10: mark(terms(X)) -> active(terms(mark(X)))
r11: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r12: mark(recip(X)) -> active(recip(mark(X)))
r13: mark(sqr(X)) -> active(sqr(mark(X)))
r14: mark(s(X)) -> active(s(X))
r15: mark(|0|()) -> active(|0|())
r16: mark(add(X1,X2)) -> active(add(mark(X1),mark(X2)))
r17: mark(dbl(X)) -> active(dbl(mark(X)))
r18: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r19: mark(nil()) -> active(nil())
r20: terms(mark(X)) -> terms(X)
r21: terms(active(X)) -> terms(X)
r22: cons(mark(X1),X2) -> cons(X1,X2)
r23: cons(X1,mark(X2)) -> cons(X1,X2)
r24: cons(active(X1),X2) -> cons(X1,X2)
r25: cons(X1,active(X2)) -> cons(X1,X2)
r26: recip(mark(X)) -> recip(X)
r27: recip(active(X)) -> recip(X)
r28: sqr(mark(X)) -> sqr(X)
r29: sqr(active(X)) -> sqr(X)
r30: s(mark(X)) -> s(X)
r31: s(active(X)) -> s(X)
r32: add(mark(X1),X2) -> add(X1,X2)
r33: add(X1,mark(X2)) -> add(X1,X2)
r34: add(active(X1),X2) -> add(X1,X2)
r35: add(X1,active(X2)) -> add(X1,X2)
r36: dbl(mark(X)) -> dbl(X)
r37: dbl(active(X)) -> dbl(X)
r38: first(mark(X1),X2) -> first(X1,X2)
r39: first(X1,mark(X2)) -> first(X1,X2)
r40: first(active(X1),X2) -> first(X1,X2)
r41: first(X1,active(X2)) -> first(X1,X2)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(first(X1,X2)) -> mark#(X2)
p2: mark#(recip(X)) -> mark#(X)
p3: mark#(sqr(X)) -> active#(sqr(mark(X)))
p4: active#(dbl(s(X))) -> mark#(s(s(dbl(X))))
p5: mark#(sqr(X)) -> mark#(X)
p6: mark#(s(X)) -> active#(s(X))
p7: mark#(add(X1,X2)) -> active#(add(mark(X1),mark(X2)))
p8: mark#(add(X1,X2)) -> mark#(X1)
p9: mark#(add(X1,X2)) -> mark#(X2)
p10: mark#(dbl(X)) -> mark#(X)

and R consists of:

r1: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
r2: active(sqr(|0|())) -> mark(|0|())
r3: active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
r4: active(dbl(|0|())) -> mark(|0|())
r5: active(dbl(s(X))) -> mark(s(s(dbl(X))))
r6: active(add(|0|(),X)) -> mark(X)
r7: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r8: active(first(|0|(),X)) -> mark(nil())
r9: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r10: mark(terms(X)) -> active(terms(mark(X)))
r11: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r12: mark(recip(X)) -> active(recip(mark(X)))
r13: mark(sqr(X)) -> active(sqr(mark(X)))
r14: mark(s(X)) -> active(s(X))
r15: mark(|0|()) -> active(|0|())
r16: mark(add(X1,X2)) -> active(add(mark(X1),mark(X2)))
r17: mark(dbl(X)) -> active(dbl(mark(X)))
r18: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r19: mark(nil()) -> active(nil())
r20: terms(mark(X)) -> terms(X)
r21: terms(active(X)) -> terms(X)
r22: cons(mark(X1),X2) -> cons(X1,X2)
r23: cons(X1,mark(X2)) -> cons(X1,X2)
r24: cons(active(X1),X2) -> cons(X1,X2)
r25: cons(X1,active(X2)) -> cons(X1,X2)
r26: recip(mark(X)) -> recip(X)
r27: recip(active(X)) -> recip(X)
r28: sqr(mark(X)) -> sqr(X)
r29: sqr(active(X)) -> sqr(X)
r30: s(mark(X)) -> s(X)
r31: s(active(X)) -> s(X)
r32: add(mark(X1),X2) -> add(X1,X2)
r33: add(X1,mark(X2)) -> add(X1,X2)
r34: add(active(X1),X2) -> add(X1,X2)
r35: add(X1,active(X2)) -> add(X1,X2)
r36: dbl(mark(X)) -> dbl(X)
r37: dbl(active(X)) -> dbl(X)
r38: first(mark(X1),X2) -> first(X1,X2)
r39: first(X1,mark(X2)) -> first(X1,X2)
r40: first(active(X1),X2) -> first(X1,X2)
r41: first(X1,active(X2)) -> first(X1,X2)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30, r31, r32, r33, r34, r35, r36, r37, r38, r39, r40, r41

Take the reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        mark#_A(x1) = max{0, x1 - 1}
        first_A(x1,x2) = max{x1 + 7, x2 + 7}
        recip_A(x1) = x1 + 8
        sqr_A(x1) = x1
        active#_A(x1) = max{0, x1 - 1}
        mark_A(x1) = x1
        dbl_A(x1) = max{2, x1}
        s_A(x1) = max{0, x1 - 4}
        add_A(x1,x2) = max{x1, x2}
        active_A(x1) = x1
        terms_A(x1) = x1 + 10
        cons_A(x1,x2) = max{x1 + 2, x2 - 4}
        |0|_A = 3
        nil_A = 0
  
    precedence:
    
      |0| > terms = cons > mark# > recip > first = add > sqr = mark = dbl = active = nil > active# = s
    
    partial status:
  
      pi(mark#) = []
      pi(first) = []
      pi(recip) = []
      pi(sqr) = []
      pi(active#) = []
      pi(mark) = []
      pi(dbl) = []
      pi(s) = []
      pi(add) = []
      pi(active) = []
      pi(terms) = []
      pi(cons) = []
      pi(|0|) = []
      pi(nil) = []

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(recip(X)) -> mark#(X)
p2: mark#(sqr(X)) -> active#(sqr(mark(X)))
p3: active#(dbl(s(X))) -> mark#(s(s(dbl(X))))
p4: mark#(sqr(X)) -> mark#(X)
p5: mark#(s(X)) -> active#(s(X))
p6: mark#(add(X1,X2)) -> active#(add(mark(X1),mark(X2)))
p7: mark#(add(X1,X2)) -> mark#(X1)
p8: mark#(add(X1,X2)) -> mark#(X2)
p9: mark#(dbl(X)) -> mark#(X)

and R consists of:

r1: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
r2: active(sqr(|0|())) -> mark(|0|())
r3: active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
r4: active(dbl(|0|())) -> mark(|0|())
r5: active(dbl(s(X))) -> mark(s(s(dbl(X))))
r6: active(add(|0|(),X)) -> mark(X)
r7: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r8: active(first(|0|(),X)) -> mark(nil())
r9: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r10: mark(terms(X)) -> active(terms(mark(X)))
r11: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r12: mark(recip(X)) -> active(recip(mark(X)))
r13: mark(sqr(X)) -> active(sqr(mark(X)))
r14: mark(s(X)) -> active(s(X))
r15: mark(|0|()) -> active(|0|())
r16: mark(add(X1,X2)) -> active(add(mark(X1),mark(X2)))
r17: mark(dbl(X)) -> active(dbl(mark(X)))
r18: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r19: mark(nil()) -> active(nil())
r20: terms(mark(X)) -> terms(X)
r21: terms(active(X)) -> terms(X)
r22: cons(mark(X1),X2) -> cons(X1,X2)
r23: cons(X1,mark(X2)) -> cons(X1,X2)
r24: cons(active(X1),X2) -> cons(X1,X2)
r25: cons(X1,active(X2)) -> cons(X1,X2)
r26: recip(mark(X)) -> recip(X)
r27: recip(active(X)) -> recip(X)
r28: sqr(mark(X)) -> sqr(X)
r29: sqr(active(X)) -> sqr(X)
r30: s(mark(X)) -> s(X)
r31: s(active(X)) -> s(X)
r32: add(mark(X1),X2) -> add(X1,X2)
r33: add(X1,mark(X2)) -> add(X1,X2)
r34: add(active(X1),X2) -> add(X1,X2)
r35: add(X1,active(X2)) -> add(X1,X2)
r36: dbl(mark(X)) -> dbl(X)
r37: dbl(active(X)) -> dbl(X)
r38: first(mark(X1),X2) -> first(X1,X2)
r39: first(X1,mark(X2)) -> first(X1,X2)
r40: first(active(X1),X2) -> first(X1,X2)
r41: first(X1,active(X2)) -> first(X1,X2)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8, p9}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(recip(X)) -> mark#(X)
p2: mark#(dbl(X)) -> mark#(X)
p3: mark#(add(X1,X2)) -> mark#(X2)
p4: mark#(add(X1,X2)) -> mark#(X1)
p5: mark#(add(X1,X2)) -> active#(add(mark(X1),mark(X2)))
p6: active#(dbl(s(X))) -> mark#(s(s(dbl(X))))
p7: mark#(s(X)) -> active#(s(X))
p8: mark#(sqr(X)) -> mark#(X)
p9: mark#(sqr(X)) -> active#(sqr(mark(X)))

and R consists of:

r1: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
r2: active(sqr(|0|())) -> mark(|0|())
r3: active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
r4: active(dbl(|0|())) -> mark(|0|())
r5: active(dbl(s(X))) -> mark(s(s(dbl(X))))
r6: active(add(|0|(),X)) -> mark(X)
r7: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r8: active(first(|0|(),X)) -> mark(nil())
r9: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r10: mark(terms(X)) -> active(terms(mark(X)))
r11: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r12: mark(recip(X)) -> active(recip(mark(X)))
r13: mark(sqr(X)) -> active(sqr(mark(X)))
r14: mark(s(X)) -> active(s(X))
r15: mark(|0|()) -> active(|0|())
r16: mark(add(X1,X2)) -> active(add(mark(X1),mark(X2)))
r17: mark(dbl(X)) -> active(dbl(mark(X)))
r18: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r19: mark(nil()) -> active(nil())
r20: terms(mark(X)) -> terms(X)
r21: terms(active(X)) -> terms(X)
r22: cons(mark(X1),X2) -> cons(X1,X2)
r23: cons(X1,mark(X2)) -> cons(X1,X2)
r24: cons(active(X1),X2) -> cons(X1,X2)
r25: cons(X1,active(X2)) -> cons(X1,X2)
r26: recip(mark(X)) -> recip(X)
r27: recip(active(X)) -> recip(X)
r28: sqr(mark(X)) -> sqr(X)
r29: sqr(active(X)) -> sqr(X)
r30: s(mark(X)) -> s(X)
r31: s(active(X)) -> s(X)
r32: add(mark(X1),X2) -> add(X1,X2)
r33: add(X1,mark(X2)) -> add(X1,X2)
r34: add(active(X1),X2) -> add(X1,X2)
r35: add(X1,active(X2)) -> add(X1,X2)
r36: dbl(mark(X)) -> dbl(X)
r37: dbl(active(X)) -> dbl(X)
r38: first(mark(X1),X2) -> first(X1,X2)
r39: first(X1,mark(X2)) -> first(X1,X2)
r40: first(active(X1),X2) -> first(X1,X2)
r41: first(X1,active(X2)) -> first(X1,X2)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30, r31, r32, r33, r34, r35, r36, r37, r38, r39, r40, r41

Take the reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        mark#_A(x1) = max{60, x1 + 47}
        recip_A(x1) = max{66, x1 + 61}
        dbl_A(x1) = max{16, x1}
        add_A(x1,x2) = max{23, x1 + 8, x2 + 5}
        active#_A(x1) = max{61, x1 + 47}
        mark_A(x1) = max{4, x1}
        s_A(x1) = 15
        sqr_A(x1) = max{22, x1}
        active_A(x1) = max{4, x1}
        terms_A(x1) = max{3, x1 - 1}
        cons_A(x1,x2) = max{2, x2 - 11}
        |0|_A = 14
        first_A(x1,x2) = 16
        nil_A = 15
  
    precedence:
    
      mark# = recip = dbl = add = active# = mark = s = sqr = active = terms = cons = |0| = first = nil
    
    partial status:
  
      pi(mark#) = []
      pi(recip) = []
      pi(dbl) = []
      pi(add) = []
      pi(active#) = []
      pi(mark) = [1]
      pi(s) = []
      pi(sqr) = []
      pi(active) = [1]
      pi(terms) = []
      pi(cons) = []
      pi(|0|) = []
      pi(first) = []
      pi(nil) = []

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(dbl(X)) -> mark#(X)
p2: mark#(add(X1,X2)) -> mark#(X2)
p3: mark#(add(X1,X2)) -> mark#(X1)
p4: mark#(add(X1,X2)) -> active#(add(mark(X1),mark(X2)))
p5: active#(dbl(s(X))) -> mark#(s(s(dbl(X))))
p6: mark#(s(X)) -> active#(s(X))
p7: mark#(sqr(X)) -> mark#(X)
p8: mark#(sqr(X)) -> active#(sqr(mark(X)))

and R consists of:

r1: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
r2: active(sqr(|0|())) -> mark(|0|())
r3: active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
r4: active(dbl(|0|())) -> mark(|0|())
r5: active(dbl(s(X))) -> mark(s(s(dbl(X))))
r6: active(add(|0|(),X)) -> mark(X)
r7: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r8: active(first(|0|(),X)) -> mark(nil())
r9: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r10: mark(terms(X)) -> active(terms(mark(X)))
r11: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r12: mark(recip(X)) -> active(recip(mark(X)))
r13: mark(sqr(X)) -> active(sqr(mark(X)))
r14: mark(s(X)) -> active(s(X))
r15: mark(|0|()) -> active(|0|())
r16: mark(add(X1,X2)) -> active(add(mark(X1),mark(X2)))
r17: mark(dbl(X)) -> active(dbl(mark(X)))
r18: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r19: mark(nil()) -> active(nil())
r20: terms(mark(X)) -> terms(X)
r21: terms(active(X)) -> terms(X)
r22: cons(mark(X1),X2) -> cons(X1,X2)
r23: cons(X1,mark(X2)) -> cons(X1,X2)
r24: cons(active(X1),X2) -> cons(X1,X2)
r25: cons(X1,active(X2)) -> cons(X1,X2)
r26: recip(mark(X)) -> recip(X)
r27: recip(active(X)) -> recip(X)
r28: sqr(mark(X)) -> sqr(X)
r29: sqr(active(X)) -> sqr(X)
r30: s(mark(X)) -> s(X)
r31: s(active(X)) -> s(X)
r32: add(mark(X1),X2) -> add(X1,X2)
r33: add(X1,mark(X2)) -> add(X1,X2)
r34: add(active(X1),X2) -> add(X1,X2)
r35: add(X1,active(X2)) -> add(X1,X2)
r36: dbl(mark(X)) -> dbl(X)
r37: dbl(active(X)) -> dbl(X)
r38: first(mark(X1),X2) -> first(X1,X2)
r39: first(X1,mark(X2)) -> first(X1,X2)
r40: first(active(X1),X2) -> first(X1,X2)
r41: first(X1,active(X2)) -> first(X1,X2)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(dbl(X)) -> mark#(X)
p2: mark#(sqr(X)) -> active#(sqr(mark(X)))
p3: active#(dbl(s(X))) -> mark#(s(s(dbl(X))))
p4: mark#(sqr(X)) -> mark#(X)
p5: mark#(s(X)) -> active#(s(X))
p6: mark#(add(X1,X2)) -> active#(add(mark(X1),mark(X2)))
p7: mark#(add(X1,X2)) -> mark#(X1)
p8: mark#(add(X1,X2)) -> mark#(X2)

and R consists of:

r1: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
r2: active(sqr(|0|())) -> mark(|0|())
r3: active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
r4: active(dbl(|0|())) -> mark(|0|())
r5: active(dbl(s(X))) -> mark(s(s(dbl(X))))
r6: active(add(|0|(),X)) -> mark(X)
r7: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r8: active(first(|0|(),X)) -> mark(nil())
r9: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r10: mark(terms(X)) -> active(terms(mark(X)))
r11: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r12: mark(recip(X)) -> active(recip(mark(X)))
r13: mark(sqr(X)) -> active(sqr(mark(X)))
r14: mark(s(X)) -> active(s(X))
r15: mark(|0|()) -> active(|0|())
r16: mark(add(X1,X2)) -> active(add(mark(X1),mark(X2)))
r17: mark(dbl(X)) -> active(dbl(mark(X)))
r18: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r19: mark(nil()) -> active(nil())
r20: terms(mark(X)) -> terms(X)
r21: terms(active(X)) -> terms(X)
r22: cons(mark(X1),X2) -> cons(X1,X2)
r23: cons(X1,mark(X2)) -> cons(X1,X2)
r24: cons(active(X1),X2) -> cons(X1,X2)
r25: cons(X1,active(X2)) -> cons(X1,X2)
r26: recip(mark(X)) -> recip(X)
r27: recip(active(X)) -> recip(X)
r28: sqr(mark(X)) -> sqr(X)
r29: sqr(active(X)) -> sqr(X)
r30: s(mark(X)) -> s(X)
r31: s(active(X)) -> s(X)
r32: add(mark(X1),X2) -> add(X1,X2)
r33: add(X1,mark(X2)) -> add(X1,X2)
r34: add(active(X1),X2) -> add(X1,X2)
r35: add(X1,active(X2)) -> add(X1,X2)
r36: dbl(mark(X)) -> dbl(X)
r37: dbl(active(X)) -> dbl(X)
r38: first(mark(X1),X2) -> first(X1,X2)
r39: first(X1,mark(X2)) -> first(X1,X2)
r40: first(active(X1),X2) -> first(X1,X2)
r41: first(X1,active(X2)) -> first(X1,X2)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30, r31, r32, r33, r34, r35, r36, r37, r38, r39, r40, r41

Take the reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        mark#_A(x1) = max{74, x1 + 63}
        dbl_A(x1) = max{22, x1 + 12}
        sqr_A(x1) = max{8, x1}
        active#_A(x1) = x1 + 51
        mark_A(x1) = max{9, x1}
        s_A(x1) = 49
        add_A(x1,x2) = max{25, x1 + 16, x2 + 5}
        active_A(x1) = max{9, x1}
        terms_A(x1) = max{72, x1 + 22}
        cons_A(x1,x2) = 21
        recip_A(x1) = max{21, x1 + 11}
        |0|_A = 0
        first_A(x1,x2) = max{5, x1 - 3}
        nil_A = 1
  
    precedence:
    
      mark = active > active# > mark# > dbl = sqr = s = add = terms = cons = recip = |0| = first = nil
    
    partial status:
  
      pi(mark#) = []
      pi(dbl) = [1]
      pi(sqr) = []
      pi(active#) = []
      pi(mark) = []
      pi(s) = []
      pi(add) = [2]
      pi(active) = []
      pi(terms) = [1]
      pi(cons) = []
      pi(recip) = [1]
      pi(|0|) = []
      pi(first) = []
      pi(nil) = []

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(dbl(X)) -> mark#(X)
p2: active#(dbl(s(X))) -> mark#(s(s(dbl(X))))
p3: mark#(sqr(X)) -> mark#(X)
p4: mark#(s(X)) -> active#(s(X))
p5: mark#(add(X1,X2)) -> active#(add(mark(X1),mark(X2)))
p6: mark#(add(X1,X2)) -> mark#(X1)
p7: mark#(add(X1,X2)) -> mark#(X2)

and R consists of:

r1: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
r2: active(sqr(|0|())) -> mark(|0|())
r3: active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
r4: active(dbl(|0|())) -> mark(|0|())
r5: active(dbl(s(X))) -> mark(s(s(dbl(X))))
r6: active(add(|0|(),X)) -> mark(X)
r7: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r8: active(first(|0|(),X)) -> mark(nil())
r9: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r10: mark(terms(X)) -> active(terms(mark(X)))
r11: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r12: mark(recip(X)) -> active(recip(mark(X)))
r13: mark(sqr(X)) -> active(sqr(mark(X)))
r14: mark(s(X)) -> active(s(X))
r15: mark(|0|()) -> active(|0|())
r16: mark(add(X1,X2)) -> active(add(mark(X1),mark(X2)))
r17: mark(dbl(X)) -> active(dbl(mark(X)))
r18: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r19: mark(nil()) -> active(nil())
r20: terms(mark(X)) -> terms(X)
r21: terms(active(X)) -> terms(X)
r22: cons(mark(X1),X2) -> cons(X1,X2)
r23: cons(X1,mark(X2)) -> cons(X1,X2)
r24: cons(active(X1),X2) -> cons(X1,X2)
r25: cons(X1,active(X2)) -> cons(X1,X2)
r26: recip(mark(X)) -> recip(X)
r27: recip(active(X)) -> recip(X)
r28: sqr(mark(X)) -> sqr(X)
r29: sqr(active(X)) -> sqr(X)
r30: s(mark(X)) -> s(X)
r31: s(active(X)) -> s(X)
r32: add(mark(X1),X2) -> add(X1,X2)
r33: add(X1,mark(X2)) -> add(X1,X2)
r34: add(active(X1),X2) -> add(X1,X2)
r35: add(X1,active(X2)) -> add(X1,X2)
r36: dbl(mark(X)) -> dbl(X)
r37: dbl(active(X)) -> dbl(X)
r38: first(mark(X1),X2) -> first(X1,X2)
r39: first(X1,mark(X2)) -> first(X1,X2)
r40: first(active(X1),X2) -> first(X1,X2)
r41: first(X1,active(X2)) -> first(X1,X2)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(dbl(X)) -> mark#(X)
p2: mark#(add(X1,X2)) -> mark#(X2)
p3: mark#(add(X1,X2)) -> mark#(X1)
p4: mark#(add(X1,X2)) -> active#(add(mark(X1),mark(X2)))
p5: active#(dbl(s(X))) -> mark#(s(s(dbl(X))))
p6: mark#(s(X)) -> active#(s(X))
p7: mark#(sqr(X)) -> mark#(X)

and R consists of:

r1: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
r2: active(sqr(|0|())) -> mark(|0|())
r3: active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
r4: active(dbl(|0|())) -> mark(|0|())
r5: active(dbl(s(X))) -> mark(s(s(dbl(X))))
r6: active(add(|0|(),X)) -> mark(X)
r7: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r8: active(first(|0|(),X)) -> mark(nil())
r9: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r10: mark(terms(X)) -> active(terms(mark(X)))
r11: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r12: mark(recip(X)) -> active(recip(mark(X)))
r13: mark(sqr(X)) -> active(sqr(mark(X)))
r14: mark(s(X)) -> active(s(X))
r15: mark(|0|()) -> active(|0|())
r16: mark(add(X1,X2)) -> active(add(mark(X1),mark(X2)))
r17: mark(dbl(X)) -> active(dbl(mark(X)))
r18: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r19: mark(nil()) -> active(nil())
r20: terms(mark(X)) -> terms(X)
r21: terms(active(X)) -> terms(X)
r22: cons(mark(X1),X2) -> cons(X1,X2)
r23: cons(X1,mark(X2)) -> cons(X1,X2)
r24: cons(active(X1),X2) -> cons(X1,X2)
r25: cons(X1,active(X2)) -> cons(X1,X2)
r26: recip(mark(X)) -> recip(X)
r27: recip(active(X)) -> recip(X)
r28: sqr(mark(X)) -> sqr(X)
r29: sqr(active(X)) -> sqr(X)
r30: s(mark(X)) -> s(X)
r31: s(active(X)) -> s(X)
r32: add(mark(X1),X2) -> add(X1,X2)
r33: add(X1,mark(X2)) -> add(X1,X2)
r34: add(active(X1),X2) -> add(X1,X2)
r35: add(X1,active(X2)) -> add(X1,X2)
r36: dbl(mark(X)) -> dbl(X)
r37: dbl(active(X)) -> dbl(X)
r38: first(mark(X1),X2) -> first(X1,X2)
r39: first(X1,mark(X2)) -> first(X1,X2)
r40: first(active(X1),X2) -> first(X1,X2)
r41: first(X1,active(X2)) -> first(X1,X2)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30, r31, r32, r33, r34, r35, r36, r37, r38, r39, r40, r41

Take the reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        mark#_A(x1) = max{7, x1 + 3}
        dbl_A(x1) = max{13, x1}
        add_A(x1,x2) = max{94, x1 + 6, x2 + 50}
        active#_A(x1) = max{8, x1 - 1}
        mark_A(x1) = max{44, x1}
        s_A(x1) = 9
        sqr_A(x1) = max{22, x1}
        active_A(x1) = max{44, x1}
        terms_A(x1) = max{2, x1 - 19}
        cons_A(x1,x2) = max{1, x1 - 109, x2 - 44}
        recip_A(x1) = max{1, x1 - 42}
        |0|_A = 0
        first_A(x1,x2) = max{152, x2 + 108}
        nil_A = 64
  
    precedence:
    
      dbl = sqr > add = active# > mark# = mark = active = terms > s = |0| > cons = recip > first = nil
    
    partial status:
  
      pi(mark#) = [1]
      pi(dbl) = []
      pi(add) = []
      pi(active#) = []
      pi(mark) = []
      pi(s) = []
      pi(sqr) = []
      pi(active) = []
      pi(terms) = []
      pi(cons) = []
      pi(recip) = []
      pi(|0|) = []
      pi(first) = []
      pi(nil) = []

The next rules are strictly ordered:

  p6

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(dbl(X)) -> mark#(X)
p2: mark#(add(X1,X2)) -> mark#(X2)
p3: mark#(add(X1,X2)) -> mark#(X1)
p4: mark#(add(X1,X2)) -> active#(add(mark(X1),mark(X2)))
p5: active#(dbl(s(X))) -> mark#(s(s(dbl(X))))
p6: mark#(sqr(X)) -> mark#(X)

and R consists of:

r1: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
r2: active(sqr(|0|())) -> mark(|0|())
r3: active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
r4: active(dbl(|0|())) -> mark(|0|())
r5: active(dbl(s(X))) -> mark(s(s(dbl(X))))
r6: active(add(|0|(),X)) -> mark(X)
r7: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r8: active(first(|0|(),X)) -> mark(nil())
r9: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r10: mark(terms(X)) -> active(terms(mark(X)))
r11: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r12: mark(recip(X)) -> active(recip(mark(X)))
r13: mark(sqr(X)) -> active(sqr(mark(X)))
r14: mark(s(X)) -> active(s(X))
r15: mark(|0|()) -> active(|0|())
r16: mark(add(X1,X2)) -> active(add(mark(X1),mark(X2)))
r17: mark(dbl(X)) -> active(dbl(mark(X)))
r18: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r19: mark(nil()) -> active(nil())
r20: terms(mark(X)) -> terms(X)
r21: terms(active(X)) -> terms(X)
r22: cons(mark(X1),X2) -> cons(X1,X2)
r23: cons(X1,mark(X2)) -> cons(X1,X2)
r24: cons(active(X1),X2) -> cons(X1,X2)
r25: cons(X1,active(X2)) -> cons(X1,X2)
r26: recip(mark(X)) -> recip(X)
r27: recip(active(X)) -> recip(X)
r28: sqr(mark(X)) -> sqr(X)
r29: sqr(active(X)) -> sqr(X)
r30: s(mark(X)) -> s(X)
r31: s(active(X)) -> s(X)
r32: add(mark(X1),X2) -> add(X1,X2)
r33: add(X1,mark(X2)) -> add(X1,X2)
r34: add(active(X1),X2) -> add(X1,X2)
r35: add(X1,active(X2)) -> add(X1,X2)
r36: dbl(mark(X)) -> dbl(X)
r37: dbl(active(X)) -> dbl(X)
r38: first(mark(X1),X2) -> first(X1,X2)
r39: first(X1,mark(X2)) -> first(X1,X2)
r40: first(active(X1),X2) -> first(X1,X2)
r41: first(X1,active(X2)) -> first(X1,X2)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(dbl(X)) -> mark#(X)
p2: mark#(sqr(X)) -> mark#(X)
p3: mark#(add(X1,X2)) -> active#(add(mark(X1),mark(X2)))
p4: active#(dbl(s(X))) -> mark#(s(s(dbl(X))))
p5: mark#(add(X1,X2)) -> mark#(X1)
p6: mark#(add(X1,X2)) -> mark#(X2)

and R consists of:

r1: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
r2: active(sqr(|0|())) -> mark(|0|())
r3: active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
r4: active(dbl(|0|())) -> mark(|0|())
r5: active(dbl(s(X))) -> mark(s(s(dbl(X))))
r6: active(add(|0|(),X)) -> mark(X)
r7: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r8: active(first(|0|(),X)) -> mark(nil())
r9: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r10: mark(terms(X)) -> active(terms(mark(X)))
r11: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r12: mark(recip(X)) -> active(recip(mark(X)))
r13: mark(sqr(X)) -> active(sqr(mark(X)))
r14: mark(s(X)) -> active(s(X))
r15: mark(|0|()) -> active(|0|())
r16: mark(add(X1,X2)) -> active(add(mark(X1),mark(X2)))
r17: mark(dbl(X)) -> active(dbl(mark(X)))
r18: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r19: mark(nil()) -> active(nil())
r20: terms(mark(X)) -> terms(X)
r21: terms(active(X)) -> terms(X)
r22: cons(mark(X1),X2) -> cons(X1,X2)
r23: cons(X1,mark(X2)) -> cons(X1,X2)
r24: cons(active(X1),X2) -> cons(X1,X2)
r25: cons(X1,active(X2)) -> cons(X1,X2)
r26: recip(mark(X)) -> recip(X)
r27: recip(active(X)) -> recip(X)
r28: sqr(mark(X)) -> sqr(X)
r29: sqr(active(X)) -> sqr(X)
r30: s(mark(X)) -> s(X)
r31: s(active(X)) -> s(X)
r32: add(mark(X1),X2) -> add(X1,X2)
r33: add(X1,mark(X2)) -> add(X1,X2)
r34: add(active(X1),X2) -> add(X1,X2)
r35: add(X1,active(X2)) -> add(X1,X2)
r36: dbl(mark(X)) -> dbl(X)
r37: dbl(active(X)) -> dbl(X)
r38: first(mark(X1),X2) -> first(X1,X2)
r39: first(X1,mark(X2)) -> first(X1,X2)
r40: first(active(X1),X2) -> first(X1,X2)
r41: first(X1,active(X2)) -> first(X1,X2)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30, r31, r32, r33, r34, r35, r36, r37, r38, r39, r40, r41

Take the reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        mark#_A(x1) = max{15, x1}
        dbl_A(x1) = max{84, x1 + 79}
        sqr_A(x1) = x1 + 71
        add_A(x1,x2) = max{x1 + 153, x2 + 153}
        active#_A(x1) = max{153, x1 - 68}
        mark_A(x1) = x1 + 66
        s_A(x1) = 14
        active_A(x1) = max{80, x1}
        terms_A(x1) = max{103, x1 + 65}
        cons_A(x1,x2) = 26
        recip_A(x1) = 15
        |0|_A = 115
        first_A(x1,x2) = 125
        nil_A = 59
  
    precedence:
    
      |0| > mark = recip = first > sqr > mark# = dbl = add = active# > nil > s = active = terms = cons
    
    partial status:
  
      pi(mark#) = [1]
      pi(dbl) = []
      pi(sqr) = [1]
      pi(add) = [1]
      pi(active#) = []
      pi(mark) = []
      pi(s) = []
      pi(active) = [1]
      pi(terms) = [1]
      pi(cons) = []
      pi(recip) = []
      pi(|0|) = []
      pi(first) = []
      pi(nil) = []

The next rules are strictly ordered:

  p3

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(dbl(X)) -> mark#(X)
p2: mark#(sqr(X)) -> mark#(X)
p3: active#(dbl(s(X))) -> mark#(s(s(dbl(X))))
p4: mark#(add(X1,X2)) -> mark#(X1)
p5: mark#(add(X1,X2)) -> mark#(X2)

and R consists of:

r1: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
r2: active(sqr(|0|())) -> mark(|0|())
r3: active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
r4: active(dbl(|0|())) -> mark(|0|())
r5: active(dbl(s(X))) -> mark(s(s(dbl(X))))
r6: active(add(|0|(),X)) -> mark(X)
r7: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r8: active(first(|0|(),X)) -> mark(nil())
r9: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r10: mark(terms(X)) -> active(terms(mark(X)))
r11: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r12: mark(recip(X)) -> active(recip(mark(X)))
r13: mark(sqr(X)) -> active(sqr(mark(X)))
r14: mark(s(X)) -> active(s(X))
r15: mark(|0|()) -> active(|0|())
r16: mark(add(X1,X2)) -> active(add(mark(X1),mark(X2)))
r17: mark(dbl(X)) -> active(dbl(mark(X)))
r18: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r19: mark(nil()) -> active(nil())
r20: terms(mark(X)) -> terms(X)
r21: terms(active(X)) -> terms(X)
r22: cons(mark(X1),X2) -> cons(X1,X2)
r23: cons(X1,mark(X2)) -> cons(X1,X2)
r24: cons(active(X1),X2) -> cons(X1,X2)
r25: cons(X1,active(X2)) -> cons(X1,X2)
r26: recip(mark(X)) -> recip(X)
r27: recip(active(X)) -> recip(X)
r28: sqr(mark(X)) -> sqr(X)
r29: sqr(active(X)) -> sqr(X)
r30: s(mark(X)) -> s(X)
r31: s(active(X)) -> s(X)
r32: add(mark(X1),X2) -> add(X1,X2)
r33: add(X1,mark(X2)) -> add(X1,X2)
r34: add(active(X1),X2) -> add(X1,X2)
r35: add(X1,active(X2)) -> add(X1,X2)
r36: dbl(mark(X)) -> dbl(X)
r37: dbl(active(X)) -> dbl(X)
r38: first(mark(X1),X2) -> first(X1,X2)
r39: first(X1,mark(X2)) -> first(X1,X2)
r40: first(active(X1),X2) -> first(X1,X2)
r41: first(X1,active(X2)) -> first(X1,X2)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p4, p5}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(dbl(X)) -> mark#(X)
p2: mark#(add(X1,X2)) -> mark#(X2)
p3: mark#(add(X1,X2)) -> mark#(X1)
p4: mark#(sqr(X)) -> mark#(X)

and R consists of:

r1: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
r2: active(sqr(|0|())) -> mark(|0|())
r3: active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
r4: active(dbl(|0|())) -> mark(|0|())
r5: active(dbl(s(X))) -> mark(s(s(dbl(X))))
r6: active(add(|0|(),X)) -> mark(X)
r7: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r8: active(first(|0|(),X)) -> mark(nil())
r9: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r10: mark(terms(X)) -> active(terms(mark(X)))
r11: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r12: mark(recip(X)) -> active(recip(mark(X)))
r13: mark(sqr(X)) -> active(sqr(mark(X)))
r14: mark(s(X)) -> active(s(X))
r15: mark(|0|()) -> active(|0|())
r16: mark(add(X1,X2)) -> active(add(mark(X1),mark(X2)))
r17: mark(dbl(X)) -> active(dbl(mark(X)))
r18: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r19: mark(nil()) -> active(nil())
r20: terms(mark(X)) -> terms(X)
r21: terms(active(X)) -> terms(X)
r22: cons(mark(X1),X2) -> cons(X1,X2)
r23: cons(X1,mark(X2)) -> cons(X1,X2)
r24: cons(active(X1),X2) -> cons(X1,X2)
r25: cons(X1,active(X2)) -> cons(X1,X2)
r26: recip(mark(X)) -> recip(X)
r27: recip(active(X)) -> recip(X)
r28: sqr(mark(X)) -> sqr(X)
r29: sqr(active(X)) -> sqr(X)
r30: s(mark(X)) -> s(X)
r31: s(active(X)) -> s(X)
r32: add(mark(X1),X2) -> add(X1,X2)
r33: add(X1,mark(X2)) -> add(X1,X2)
r34: add(active(X1),X2) -> add(X1,X2)
r35: add(X1,active(X2)) -> add(X1,X2)
r36: dbl(mark(X)) -> dbl(X)
r37: dbl(active(X)) -> dbl(X)
r38: first(mark(X1),X2) -> first(X1,X2)
r39: first(X1,mark(X2)) -> first(X1,X2)
r40: first(active(X1),X2) -> first(X1,X2)
r41: first(X1,active(X2)) -> first(X1,X2)

The set of usable rules consists of

  (no rules)

Take the monotone reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        mark#_A(x1) = max{3, x1 + 2}
        dbl_A(x1) = x1
        add_A(x1,x2) = max{x1, x2}
        sqr_A(x1) = x1 + 1
  
    precedence:
    
      mark# = dbl = add = sqr
    
    partial status:
  
      pi(mark#) = [1]
      pi(dbl) = [1]
      pi(add) = [1, 2]
      pi(sqr) = [1]

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(add(X1,X2)) -> mark#(X2)
p2: mark#(add(X1,X2)) -> mark#(X1)
p3: mark#(sqr(X)) -> mark#(X)

and R consists of:

r1: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
r2: active(sqr(|0|())) -> mark(|0|())
r3: active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
r4: active(dbl(|0|())) -> mark(|0|())
r5: active(dbl(s(X))) -> mark(s(s(dbl(X))))
r6: active(add(|0|(),X)) -> mark(X)
r7: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r8: active(first(|0|(),X)) -> mark(nil())
r9: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r10: mark(terms(X)) -> active(terms(mark(X)))
r11: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r12: mark(recip(X)) -> active(recip(mark(X)))
r13: mark(sqr(X)) -> active(sqr(mark(X)))
r14: mark(s(X)) -> active(s(X))
r15: mark(|0|()) -> active(|0|())
r16: mark(add(X1,X2)) -> active(add(mark(X1),mark(X2)))
r17: mark(dbl(X)) -> active(dbl(mark(X)))
r18: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r19: mark(nil()) -> active(nil())
r20: terms(mark(X)) -> terms(X)
r21: terms(active(X)) -> terms(X)
r22: cons(mark(X1),X2) -> cons(X1,X2)
r23: cons(X1,mark(X2)) -> cons(X1,X2)
r24: cons(active(X1),X2) -> cons(X1,X2)
r25: cons(X1,active(X2)) -> cons(X1,X2)
r26: recip(mark(X)) -> recip(X)
r27: recip(active(X)) -> recip(X)
r28: sqr(mark(X)) -> sqr(X)
r29: sqr(active(X)) -> sqr(X)
r30: s(mark(X)) -> s(X)
r31: s(active(X)) -> s(X)
r32: add(mark(X1),X2) -> add(X1,X2)
r33: add(X1,mark(X2)) -> add(X1,X2)
r34: add(active(X1),X2) -> add(X1,X2)
r35: add(X1,active(X2)) -> add(X1,X2)
r36: dbl(mark(X)) -> dbl(X)
r37: dbl(active(X)) -> dbl(X)
r38: first(mark(X1),X2) -> first(X1,X2)
r39: first(X1,mark(X2)) -> first(X1,X2)
r40: first(active(X1),X2) -> first(X1,X2)
r41: first(X1,active(X2)) -> first(X1,X2)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(add(X1,X2)) -> mark#(X2)
p2: mark#(sqr(X)) -> mark#(X)
p3: mark#(add(X1,X2)) -> mark#(X1)

and R consists of:

r1: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
r2: active(sqr(|0|())) -> mark(|0|())
r3: active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
r4: active(dbl(|0|())) -> mark(|0|())
r5: active(dbl(s(X))) -> mark(s(s(dbl(X))))
r6: active(add(|0|(),X)) -> mark(X)
r7: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r8: active(first(|0|(),X)) -> mark(nil())
r9: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r10: mark(terms(X)) -> active(terms(mark(X)))
r11: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r12: mark(recip(X)) -> active(recip(mark(X)))
r13: mark(sqr(X)) -> active(sqr(mark(X)))
r14: mark(s(X)) -> active(s(X))
r15: mark(|0|()) -> active(|0|())
r16: mark(add(X1,X2)) -> active(add(mark(X1),mark(X2)))
r17: mark(dbl(X)) -> active(dbl(mark(X)))
r18: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r19: mark(nil()) -> active(nil())
r20: terms(mark(X)) -> terms(X)
r21: terms(active(X)) -> terms(X)
r22: cons(mark(X1),X2) -> cons(X1,X2)
r23: cons(X1,mark(X2)) -> cons(X1,X2)
r24: cons(active(X1),X2) -> cons(X1,X2)
r25: cons(X1,active(X2)) -> cons(X1,X2)
r26: recip(mark(X)) -> recip(X)
r27: recip(active(X)) -> recip(X)
r28: sqr(mark(X)) -> sqr(X)
r29: sqr(active(X)) -> sqr(X)
r30: s(mark(X)) -> s(X)
r31: s(active(X)) -> s(X)
r32: add(mark(X1),X2) -> add(X1,X2)
r33: add(X1,mark(X2)) -> add(X1,X2)
r34: add(active(X1),X2) -> add(X1,X2)
r35: add(X1,active(X2)) -> add(X1,X2)
r36: dbl(mark(X)) -> dbl(X)
r37: dbl(active(X)) -> dbl(X)
r38: first(mark(X1),X2) -> first(X1,X2)
r39: first(X1,mark(X2)) -> first(X1,X2)
r40: first(active(X1),X2) -> first(X1,X2)
r41: first(X1,active(X2)) -> first(X1,X2)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        mark#_A(x1) = x1 + 2
        add_A(x1,x2) = max{x1 + 1, x2 + 1}
        sqr_A(x1) = x1
  
    precedence:
    
      mark# = add = sqr
    
    partial status:
  
      pi(mark#) = []
      pi(add) = [2]
      pi(sqr) = [1]

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(sqr(X)) -> mark#(X)
p2: mark#(add(X1,X2)) -> mark#(X1)

and R consists of:

r1: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
r2: active(sqr(|0|())) -> mark(|0|())
r3: active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
r4: active(dbl(|0|())) -> mark(|0|())
r5: active(dbl(s(X))) -> mark(s(s(dbl(X))))
r6: active(add(|0|(),X)) -> mark(X)
r7: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r8: active(first(|0|(),X)) -> mark(nil())
r9: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r10: mark(terms(X)) -> active(terms(mark(X)))
r11: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r12: mark(recip(X)) -> active(recip(mark(X)))
r13: mark(sqr(X)) -> active(sqr(mark(X)))
r14: mark(s(X)) -> active(s(X))
r15: mark(|0|()) -> active(|0|())
r16: mark(add(X1,X2)) -> active(add(mark(X1),mark(X2)))
r17: mark(dbl(X)) -> active(dbl(mark(X)))
r18: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r19: mark(nil()) -> active(nil())
r20: terms(mark(X)) -> terms(X)
r21: terms(active(X)) -> terms(X)
r22: cons(mark(X1),X2) -> cons(X1,X2)
r23: cons(X1,mark(X2)) -> cons(X1,X2)
r24: cons(active(X1),X2) -> cons(X1,X2)
r25: cons(X1,active(X2)) -> cons(X1,X2)
r26: recip(mark(X)) -> recip(X)
r27: recip(active(X)) -> recip(X)
r28: sqr(mark(X)) -> sqr(X)
r29: sqr(active(X)) -> sqr(X)
r30: s(mark(X)) -> s(X)
r31: s(active(X)) -> s(X)
r32: add(mark(X1),X2) -> add(X1,X2)
r33: add(X1,mark(X2)) -> add(X1,X2)
r34: add(active(X1),X2) -> add(X1,X2)
r35: add(X1,active(X2)) -> add(X1,X2)
r36: dbl(mark(X)) -> dbl(X)
r37: dbl(active(X)) -> dbl(X)
r38: first(mark(X1),X2) -> first(X1,X2)
r39: first(X1,mark(X2)) -> first(X1,X2)
r40: first(active(X1),X2) -> first(X1,X2)
r41: first(X1,active(X2)) -> first(X1,X2)

The estimated dependency graph contains the following SCCs:

  {p1, p2}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(sqr(X)) -> mark#(X)
p2: mark#(add(X1,X2)) -> mark#(X1)

and R consists of:

r1: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
r2: active(sqr(|0|())) -> mark(|0|())
r3: active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
r4: active(dbl(|0|())) -> mark(|0|())
r5: active(dbl(s(X))) -> mark(s(s(dbl(X))))
r6: active(add(|0|(),X)) -> mark(X)
r7: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r8: active(first(|0|(),X)) -> mark(nil())
r9: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r10: mark(terms(X)) -> active(terms(mark(X)))
r11: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r12: mark(recip(X)) -> active(recip(mark(X)))
r13: mark(sqr(X)) -> active(sqr(mark(X)))
r14: mark(s(X)) -> active(s(X))
r15: mark(|0|()) -> active(|0|())
r16: mark(add(X1,X2)) -> active(add(mark(X1),mark(X2)))
r17: mark(dbl(X)) -> active(dbl(mark(X)))
r18: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r19: mark(nil()) -> active(nil())
r20: terms(mark(X)) -> terms(X)
r21: terms(active(X)) -> terms(X)
r22: cons(mark(X1),X2) -> cons(X1,X2)
r23: cons(X1,mark(X2)) -> cons(X1,X2)
r24: cons(active(X1),X2) -> cons(X1,X2)
r25: cons(X1,active(X2)) -> cons(X1,X2)
r26: recip(mark(X)) -> recip(X)
r27: recip(active(X)) -> recip(X)
r28: sqr(mark(X)) -> sqr(X)
r29: sqr(active(X)) -> sqr(X)
r30: s(mark(X)) -> s(X)
r31: s(active(X)) -> s(X)
r32: add(mark(X1),X2) -> add(X1,X2)
r33: add(X1,mark(X2)) -> add(X1,X2)
r34: add(active(X1),X2) -> add(X1,X2)
r35: add(X1,active(X2)) -> add(X1,X2)
r36: dbl(mark(X)) -> dbl(X)
r37: dbl(active(X)) -> dbl(X)
r38: first(mark(X1),X2) -> first(X1,X2)
r39: first(X1,mark(X2)) -> first(X1,X2)
r40: first(active(X1),X2) -> first(X1,X2)
r41: first(X1,active(X2)) -> first(X1,X2)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        mark#_A(x1) = max{4, x1 + 3}
        sqr_A(x1) = x1 + 2
        add_A(x1,x2) = max{x1 + 1, x2 + 1}
  
    precedence:
    
      mark# = sqr = add
    
    partial status:
  
      pi(mark#) = []
      pi(sqr) = [1]
      pi(add) = [2]

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(add(X1,X2)) -> mark#(X1)

and R consists of:

r1: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
r2: active(sqr(|0|())) -> mark(|0|())
r3: active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
r4: active(dbl(|0|())) -> mark(|0|())
r5: active(dbl(s(X))) -> mark(s(s(dbl(X))))
r6: active(add(|0|(),X)) -> mark(X)
r7: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r8: active(first(|0|(),X)) -> mark(nil())
r9: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r10: mark(terms(X)) -> active(terms(mark(X)))
r11: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r12: mark(recip(X)) -> active(recip(mark(X)))
r13: mark(sqr(X)) -> active(sqr(mark(X)))
r14: mark(s(X)) -> active(s(X))
r15: mark(|0|()) -> active(|0|())
r16: mark(add(X1,X2)) -> active(add(mark(X1),mark(X2)))
r17: mark(dbl(X)) -> active(dbl(mark(X)))
r18: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r19: mark(nil()) -> active(nil())
r20: terms(mark(X)) -> terms(X)
r21: terms(active(X)) -> terms(X)
r22: cons(mark(X1),X2) -> cons(X1,X2)
r23: cons(X1,mark(X2)) -> cons(X1,X2)
r24: cons(active(X1),X2) -> cons(X1,X2)
r25: cons(X1,active(X2)) -> cons(X1,X2)
r26: recip(mark(X)) -> recip(X)
r27: recip(active(X)) -> recip(X)
r28: sqr(mark(X)) -> sqr(X)
r29: sqr(active(X)) -> sqr(X)
r30: s(mark(X)) -> s(X)
r31: s(active(X)) -> s(X)
r32: add(mark(X1),X2) -> add(X1,X2)
r33: add(X1,mark(X2)) -> add(X1,X2)
r34: add(active(X1),X2) -> add(X1,X2)
r35: add(X1,active(X2)) -> add(X1,X2)
r36: dbl(mark(X)) -> dbl(X)
r37: dbl(active(X)) -> dbl(X)
r38: first(mark(X1),X2) -> first(X1,X2)
r39: first(X1,mark(X2)) -> first(X1,X2)
r40: first(active(X1),X2) -> first(X1,X2)
r41: first(X1,active(X2)) -> first(X1,X2)

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(add(X1,X2)) -> mark#(X1)

and R consists of:

r1: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
r2: active(sqr(|0|())) -> mark(|0|())
r3: active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
r4: active(dbl(|0|())) -> mark(|0|())
r5: active(dbl(s(X))) -> mark(s(s(dbl(X))))
r6: active(add(|0|(),X)) -> mark(X)
r7: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r8: active(first(|0|(),X)) -> mark(nil())
r9: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r10: mark(terms(X)) -> active(terms(mark(X)))
r11: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r12: mark(recip(X)) -> active(recip(mark(X)))
r13: mark(sqr(X)) -> active(sqr(mark(X)))
r14: mark(s(X)) -> active(s(X))
r15: mark(|0|()) -> active(|0|())
r16: mark(add(X1,X2)) -> active(add(mark(X1),mark(X2)))
r17: mark(dbl(X)) -> active(dbl(mark(X)))
r18: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r19: mark(nil()) -> active(nil())
r20: terms(mark(X)) -> terms(X)
r21: terms(active(X)) -> terms(X)
r22: cons(mark(X1),X2) -> cons(X1,X2)
r23: cons(X1,mark(X2)) -> cons(X1,X2)
r24: cons(active(X1),X2) -> cons(X1,X2)
r25: cons(X1,active(X2)) -> cons(X1,X2)
r26: recip(mark(X)) -> recip(X)
r27: recip(active(X)) -> recip(X)
r28: sqr(mark(X)) -> sqr(X)
r29: sqr(active(X)) -> sqr(X)
r30: s(mark(X)) -> s(X)
r31: s(active(X)) -> s(X)
r32: add(mark(X1),X2) -> add(X1,X2)
r33: add(X1,mark(X2)) -> add(X1,X2)
r34: add(active(X1),X2) -> add(X1,X2)
r35: add(X1,active(X2)) -> add(X1,X2)
r36: dbl(mark(X)) -> dbl(X)
r37: dbl(active(X)) -> dbl(X)
r38: first(mark(X1),X2) -> first(X1,X2)
r39: first(X1,mark(X2)) -> first(X1,X2)
r40: first(active(X1),X2) -> first(X1,X2)
r41: first(X1,active(X2)) -> first(X1,X2)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        mark#_A(x1) = x1 + 2
        add_A(x1,x2) = max{x1 + 1, x2 + 1}
  
    precedence:
    
      mark# = add
    
    partial status:
  
      pi(mark#) = []
      pi(add) = [2]

The next rules are strictly ordered:

  p1

We remove them from the problem.  Then no dependency pair remains.

-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: cons#(mark(X1),X2) -> cons#(X1,X2)
p2: cons#(X1,active(X2)) -> cons#(X1,X2)
p3: cons#(active(X1),X2) -> cons#(X1,X2)
p4: cons#(X1,mark(X2)) -> cons#(X1,X2)

and R consists of:

r1: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
r2: active(sqr(|0|())) -> mark(|0|())
r3: active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
r4: active(dbl(|0|())) -> mark(|0|())
r5: active(dbl(s(X))) -> mark(s(s(dbl(X))))
r6: active(add(|0|(),X)) -> mark(X)
r7: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r8: active(first(|0|(),X)) -> mark(nil())
r9: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r10: mark(terms(X)) -> active(terms(mark(X)))
r11: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r12: mark(recip(X)) -> active(recip(mark(X)))
r13: mark(sqr(X)) -> active(sqr(mark(X)))
r14: mark(s(X)) -> active(s(X))
r15: mark(|0|()) -> active(|0|())
r16: mark(add(X1,X2)) -> active(add(mark(X1),mark(X2)))
r17: mark(dbl(X)) -> active(dbl(mark(X)))
r18: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r19: mark(nil()) -> active(nil())
r20: terms(mark(X)) -> terms(X)
r21: terms(active(X)) -> terms(X)
r22: cons(mark(X1),X2) -> cons(X1,X2)
r23: cons(X1,mark(X2)) -> cons(X1,X2)
r24: cons(active(X1),X2) -> cons(X1,X2)
r25: cons(X1,active(X2)) -> cons(X1,X2)
r26: recip(mark(X)) -> recip(X)
r27: recip(active(X)) -> recip(X)
r28: sqr(mark(X)) -> sqr(X)
r29: sqr(active(X)) -> sqr(X)
r30: s(mark(X)) -> s(X)
r31: s(active(X)) -> s(X)
r32: add(mark(X1),X2) -> add(X1,X2)
r33: add(X1,mark(X2)) -> add(X1,X2)
r34: add(active(X1),X2) -> add(X1,X2)
r35: add(X1,active(X2)) -> add(X1,X2)
r36: dbl(mark(X)) -> dbl(X)
r37: dbl(active(X)) -> dbl(X)
r38: first(mark(X1),X2) -> first(X1,X2)
r39: first(X1,mark(X2)) -> first(X1,X2)
r40: first(active(X1),X2) -> first(X1,X2)
r41: first(X1,active(X2)) -> first(X1,X2)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        cons#_A(x1,x2) = max{0, x2 - 2}
        mark_A(x1) = x1
        active_A(x1) = max{3, x1 + 1}
  
    precedence:
    
      cons# = mark = active
    
    partial status:
  
      pi(cons#) = []
      pi(mark) = [1]
      pi(active) = [1]

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: cons#(mark(X1),X2) -> cons#(X1,X2)
p2: cons#(active(X1),X2) -> cons#(X1,X2)
p3: cons#(X1,mark(X2)) -> cons#(X1,X2)

and R consists of:

r1: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
r2: active(sqr(|0|())) -> mark(|0|())
r3: active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
r4: active(dbl(|0|())) -> mark(|0|())
r5: active(dbl(s(X))) -> mark(s(s(dbl(X))))
r6: active(add(|0|(),X)) -> mark(X)
r7: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r8: active(first(|0|(),X)) -> mark(nil())
r9: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r10: mark(terms(X)) -> active(terms(mark(X)))
r11: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r12: mark(recip(X)) -> active(recip(mark(X)))
r13: mark(sqr(X)) -> active(sqr(mark(X)))
r14: mark(s(X)) -> active(s(X))
r15: mark(|0|()) -> active(|0|())
r16: mark(add(X1,X2)) -> active(add(mark(X1),mark(X2)))
r17: mark(dbl(X)) -> active(dbl(mark(X)))
r18: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r19: mark(nil()) -> active(nil())
r20: terms(mark(X)) -> terms(X)
r21: terms(active(X)) -> terms(X)
r22: cons(mark(X1),X2) -> cons(X1,X2)
r23: cons(X1,mark(X2)) -> cons(X1,X2)
r24: cons(active(X1),X2) -> cons(X1,X2)
r25: cons(X1,active(X2)) -> cons(X1,X2)
r26: recip(mark(X)) -> recip(X)
r27: recip(active(X)) -> recip(X)
r28: sqr(mark(X)) -> sqr(X)
r29: sqr(active(X)) -> sqr(X)
r30: s(mark(X)) -> s(X)
r31: s(active(X)) -> s(X)
r32: add(mark(X1),X2) -> add(X1,X2)
r33: add(X1,mark(X2)) -> add(X1,X2)
r34: add(active(X1),X2) -> add(X1,X2)
r35: add(X1,active(X2)) -> add(X1,X2)
r36: dbl(mark(X)) -> dbl(X)
r37: dbl(active(X)) -> dbl(X)
r38: first(mark(X1),X2) -> first(X1,X2)
r39: first(X1,mark(X2)) -> first(X1,X2)
r40: first(active(X1),X2) -> first(X1,X2)
r41: first(X1,active(X2)) -> first(X1,X2)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: cons#(mark(X1),X2) -> cons#(X1,X2)
p2: cons#(X1,mark(X2)) -> cons#(X1,X2)
p3: cons#(active(X1),X2) -> cons#(X1,X2)

and R consists of:

r1: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
r2: active(sqr(|0|())) -> mark(|0|())
r3: active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
r4: active(dbl(|0|())) -> mark(|0|())
r5: active(dbl(s(X))) -> mark(s(s(dbl(X))))
r6: active(add(|0|(),X)) -> mark(X)
r7: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r8: active(first(|0|(),X)) -> mark(nil())
r9: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r10: mark(terms(X)) -> active(terms(mark(X)))
r11: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r12: mark(recip(X)) -> active(recip(mark(X)))
r13: mark(sqr(X)) -> active(sqr(mark(X)))
r14: mark(s(X)) -> active(s(X))
r15: mark(|0|()) -> active(|0|())
r16: mark(add(X1,X2)) -> active(add(mark(X1),mark(X2)))
r17: mark(dbl(X)) -> active(dbl(mark(X)))
r18: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r19: mark(nil()) -> active(nil())
r20: terms(mark(X)) -> terms(X)
r21: terms(active(X)) -> terms(X)
r22: cons(mark(X1),X2) -> cons(X1,X2)
r23: cons(X1,mark(X2)) -> cons(X1,X2)
r24: cons(active(X1),X2) -> cons(X1,X2)
r25: cons(X1,active(X2)) -> cons(X1,X2)
r26: recip(mark(X)) -> recip(X)
r27: recip(active(X)) -> recip(X)
r28: sqr(mark(X)) -> sqr(X)
r29: sqr(active(X)) -> sqr(X)
r30: s(mark(X)) -> s(X)
r31: s(active(X)) -> s(X)
r32: add(mark(X1),X2) -> add(X1,X2)
r33: add(X1,mark(X2)) -> add(X1,X2)
r34: add(active(X1),X2) -> add(X1,X2)
r35: add(X1,active(X2)) -> add(X1,X2)
r36: dbl(mark(X)) -> dbl(X)
r37: dbl(active(X)) -> dbl(X)
r38: first(mark(X1),X2) -> first(X1,X2)
r39: first(X1,mark(X2)) -> first(X1,X2)
r40: first(active(X1),X2) -> first(X1,X2)
r41: first(X1,active(X2)) -> first(X1,X2)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        cons#_A(x1,x2) = max{0, x1 - 2}
        mark_A(x1) = max{3, x1 + 1}
        active_A(x1) = x1
  
    precedence:
    
      cons# = mark = active
    
    partial status:
  
      pi(cons#) = []
      pi(mark) = [1]
      pi(active) = [1]

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: cons#(X1,mark(X2)) -> cons#(X1,X2)
p2: cons#(active(X1),X2) -> cons#(X1,X2)

and R consists of:

r1: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
r2: active(sqr(|0|())) -> mark(|0|())
r3: active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
r4: active(dbl(|0|())) -> mark(|0|())
r5: active(dbl(s(X))) -> mark(s(s(dbl(X))))
r6: active(add(|0|(),X)) -> mark(X)
r7: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r8: active(first(|0|(),X)) -> mark(nil())
r9: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r10: mark(terms(X)) -> active(terms(mark(X)))
r11: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r12: mark(recip(X)) -> active(recip(mark(X)))
r13: mark(sqr(X)) -> active(sqr(mark(X)))
r14: mark(s(X)) -> active(s(X))
r15: mark(|0|()) -> active(|0|())
r16: mark(add(X1,X2)) -> active(add(mark(X1),mark(X2)))
r17: mark(dbl(X)) -> active(dbl(mark(X)))
r18: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r19: mark(nil()) -> active(nil())
r20: terms(mark(X)) -> terms(X)
r21: terms(active(X)) -> terms(X)
r22: cons(mark(X1),X2) -> cons(X1,X2)
r23: cons(X1,mark(X2)) -> cons(X1,X2)
r24: cons(active(X1),X2) -> cons(X1,X2)
r25: cons(X1,active(X2)) -> cons(X1,X2)
r26: recip(mark(X)) -> recip(X)
r27: recip(active(X)) -> recip(X)
r28: sqr(mark(X)) -> sqr(X)
r29: sqr(active(X)) -> sqr(X)
r30: s(mark(X)) -> s(X)
r31: s(active(X)) -> s(X)
r32: add(mark(X1),X2) -> add(X1,X2)
r33: add(X1,mark(X2)) -> add(X1,X2)
r34: add(active(X1),X2) -> add(X1,X2)
r35: add(X1,active(X2)) -> add(X1,X2)
r36: dbl(mark(X)) -> dbl(X)
r37: dbl(active(X)) -> dbl(X)
r38: first(mark(X1),X2) -> first(X1,X2)
r39: first(X1,mark(X2)) -> first(X1,X2)
r40: first(active(X1),X2) -> first(X1,X2)
r41: first(X1,active(X2)) -> first(X1,X2)

The estimated dependency graph contains the following SCCs:

  {p1, p2}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: cons#(X1,mark(X2)) -> cons#(X1,X2)
p2: cons#(active(X1),X2) -> cons#(X1,X2)

and R consists of:

r1: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
r2: active(sqr(|0|())) -> mark(|0|())
r3: active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
r4: active(dbl(|0|())) -> mark(|0|())
r5: active(dbl(s(X))) -> mark(s(s(dbl(X))))
r6: active(add(|0|(),X)) -> mark(X)
r7: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r8: active(first(|0|(),X)) -> mark(nil())
r9: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r10: mark(terms(X)) -> active(terms(mark(X)))
r11: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r12: mark(recip(X)) -> active(recip(mark(X)))
r13: mark(sqr(X)) -> active(sqr(mark(X)))
r14: mark(s(X)) -> active(s(X))
r15: mark(|0|()) -> active(|0|())
r16: mark(add(X1,X2)) -> active(add(mark(X1),mark(X2)))
r17: mark(dbl(X)) -> active(dbl(mark(X)))
r18: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r19: mark(nil()) -> active(nil())
r20: terms(mark(X)) -> terms(X)
r21: terms(active(X)) -> terms(X)
r22: cons(mark(X1),X2) -> cons(X1,X2)
r23: cons(X1,mark(X2)) -> cons(X1,X2)
r24: cons(active(X1),X2) -> cons(X1,X2)
r25: cons(X1,active(X2)) -> cons(X1,X2)
r26: recip(mark(X)) -> recip(X)
r27: recip(active(X)) -> recip(X)
r28: sqr(mark(X)) -> sqr(X)
r29: sqr(active(X)) -> sqr(X)
r30: s(mark(X)) -> s(X)
r31: s(active(X)) -> s(X)
r32: add(mark(X1),X2) -> add(X1,X2)
r33: add(X1,mark(X2)) -> add(X1,X2)
r34: add(active(X1),X2) -> add(X1,X2)
r35: add(X1,active(X2)) -> add(X1,X2)
r36: dbl(mark(X)) -> dbl(X)
r37: dbl(active(X)) -> dbl(X)
r38: first(mark(X1),X2) -> first(X1,X2)
r39: first(X1,mark(X2)) -> first(X1,X2)
r40: first(active(X1),X2) -> first(X1,X2)
r41: first(X1,active(X2)) -> first(X1,X2)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        cons#_A(x1,x2) = x2 + 1
        mark_A(x1) = x1 + 1
        active_A(x1) = x1 + 1
  
    precedence:
    
      cons# = mark = active
    
    partial status:
  
      pi(cons#) = []
      pi(mark) = [1]
      pi(active) = [1]

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: cons#(active(X1),X2) -> cons#(X1,X2)

and R consists of:

r1: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
r2: active(sqr(|0|())) -> mark(|0|())
r3: active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
r4: active(dbl(|0|())) -> mark(|0|())
r5: active(dbl(s(X))) -> mark(s(s(dbl(X))))
r6: active(add(|0|(),X)) -> mark(X)
r7: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r8: active(first(|0|(),X)) -> mark(nil())
r9: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r10: mark(terms(X)) -> active(terms(mark(X)))
r11: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r12: mark(recip(X)) -> active(recip(mark(X)))
r13: mark(sqr(X)) -> active(sqr(mark(X)))
r14: mark(s(X)) -> active(s(X))
r15: mark(|0|()) -> active(|0|())
r16: mark(add(X1,X2)) -> active(add(mark(X1),mark(X2)))
r17: mark(dbl(X)) -> active(dbl(mark(X)))
r18: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r19: mark(nil()) -> active(nil())
r20: terms(mark(X)) -> terms(X)
r21: terms(active(X)) -> terms(X)
r22: cons(mark(X1),X2) -> cons(X1,X2)
r23: cons(X1,mark(X2)) -> cons(X1,X2)
r24: cons(active(X1),X2) -> cons(X1,X2)
r25: cons(X1,active(X2)) -> cons(X1,X2)
r26: recip(mark(X)) -> recip(X)
r27: recip(active(X)) -> recip(X)
r28: sqr(mark(X)) -> sqr(X)
r29: sqr(active(X)) -> sqr(X)
r30: s(mark(X)) -> s(X)
r31: s(active(X)) -> s(X)
r32: add(mark(X1),X2) -> add(X1,X2)
r33: add(X1,mark(X2)) -> add(X1,X2)
r34: add(active(X1),X2) -> add(X1,X2)
r35: add(X1,active(X2)) -> add(X1,X2)
r36: dbl(mark(X)) -> dbl(X)
r37: dbl(active(X)) -> dbl(X)
r38: first(mark(X1),X2) -> first(X1,X2)
r39: first(X1,mark(X2)) -> first(X1,X2)
r40: first(active(X1),X2) -> first(X1,X2)
r41: first(X1,active(X2)) -> first(X1,X2)

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: cons#(active(X1),X2) -> cons#(X1,X2)

and R consists of:

r1: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
r2: active(sqr(|0|())) -> mark(|0|())
r3: active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
r4: active(dbl(|0|())) -> mark(|0|())
r5: active(dbl(s(X))) -> mark(s(s(dbl(X))))
r6: active(add(|0|(),X)) -> mark(X)
r7: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r8: active(first(|0|(),X)) -> mark(nil())
r9: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r10: mark(terms(X)) -> active(terms(mark(X)))
r11: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r12: mark(recip(X)) -> active(recip(mark(X)))
r13: mark(sqr(X)) -> active(sqr(mark(X)))
r14: mark(s(X)) -> active(s(X))
r15: mark(|0|()) -> active(|0|())
r16: mark(add(X1,X2)) -> active(add(mark(X1),mark(X2)))
r17: mark(dbl(X)) -> active(dbl(mark(X)))
r18: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r19: mark(nil()) -> active(nil())
r20: terms(mark(X)) -> terms(X)
r21: terms(active(X)) -> terms(X)
r22: cons(mark(X1),X2) -> cons(X1,X2)
r23: cons(X1,mark(X2)) -> cons(X1,X2)
r24: cons(active(X1),X2) -> cons(X1,X2)
r25: cons(X1,active(X2)) -> cons(X1,X2)
r26: recip(mark(X)) -> recip(X)
r27: recip(active(X)) -> recip(X)
r28: sqr(mark(X)) -> sqr(X)
r29: sqr(active(X)) -> sqr(X)
r30: s(mark(X)) -> s(X)
r31: s(active(X)) -> s(X)
r32: add(mark(X1),X2) -> add(X1,X2)
r33: add(X1,mark(X2)) -> add(X1,X2)
r34: add(active(X1),X2) -> add(X1,X2)
r35: add(X1,active(X2)) -> add(X1,X2)
r36: dbl(mark(X)) -> dbl(X)
r37: dbl(active(X)) -> dbl(X)
r38: first(mark(X1),X2) -> first(X1,X2)
r39: first(X1,mark(X2)) -> first(X1,X2)
r40: first(active(X1),X2) -> first(X1,X2)
r41: first(X1,active(X2)) -> first(X1,X2)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        cons#_A(x1,x2) = max{1, x1, x2}
        active_A(x1) = x1 + 1
  
    precedence:
    
      cons# = active
    
    partial status:
  
      pi(cons#) = [1, 2]
      pi(active) = []

The next rules are strictly ordered:

  p1

We remove them from the problem.  Then no dependency pair remains.

-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: recip#(mark(X)) -> recip#(X)
p2: recip#(active(X)) -> recip#(X)

and R consists of:

r1: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
r2: active(sqr(|0|())) -> mark(|0|())
r3: active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
r4: active(dbl(|0|())) -> mark(|0|())
r5: active(dbl(s(X))) -> mark(s(s(dbl(X))))
r6: active(add(|0|(),X)) -> mark(X)
r7: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r8: active(first(|0|(),X)) -> mark(nil())
r9: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r10: mark(terms(X)) -> active(terms(mark(X)))
r11: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r12: mark(recip(X)) -> active(recip(mark(X)))
r13: mark(sqr(X)) -> active(sqr(mark(X)))
r14: mark(s(X)) -> active(s(X))
r15: mark(|0|()) -> active(|0|())
r16: mark(add(X1,X2)) -> active(add(mark(X1),mark(X2)))
r17: mark(dbl(X)) -> active(dbl(mark(X)))
r18: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r19: mark(nil()) -> active(nil())
r20: terms(mark(X)) -> terms(X)
r21: terms(active(X)) -> terms(X)
r22: cons(mark(X1),X2) -> cons(X1,X2)
r23: cons(X1,mark(X2)) -> cons(X1,X2)
r24: cons(active(X1),X2) -> cons(X1,X2)
r25: cons(X1,active(X2)) -> cons(X1,X2)
r26: recip(mark(X)) -> recip(X)
r27: recip(active(X)) -> recip(X)
r28: sqr(mark(X)) -> sqr(X)
r29: sqr(active(X)) -> sqr(X)
r30: s(mark(X)) -> s(X)
r31: s(active(X)) -> s(X)
r32: add(mark(X1),X2) -> add(X1,X2)
r33: add(X1,mark(X2)) -> add(X1,X2)
r34: add(active(X1),X2) -> add(X1,X2)
r35: add(X1,active(X2)) -> add(X1,X2)
r36: dbl(mark(X)) -> dbl(X)
r37: dbl(active(X)) -> dbl(X)
r38: first(mark(X1),X2) -> first(X1,X2)
r39: first(X1,mark(X2)) -> first(X1,X2)
r40: first(active(X1),X2) -> first(X1,X2)
r41: first(X1,active(X2)) -> first(X1,X2)

The set of usable rules consists of

  (no rules)

Take the monotone reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        recip#_A(x1) = max{3, x1 + 2}
        mark_A(x1) = x1
        active_A(x1) = x1 + 1
  
    precedence:
    
      recip# = mark = active
    
    partial status:
  
      pi(recip#) = [1]
      pi(mark) = [1]
      pi(active) = [1]

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: recip#(active(X)) -> recip#(X)

and R consists of:

r1: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
r2: active(sqr(|0|())) -> mark(|0|())
r3: active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
r4: active(dbl(|0|())) -> mark(|0|())
r5: active(dbl(s(X))) -> mark(s(s(dbl(X))))
r6: active(add(|0|(),X)) -> mark(X)
r7: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r8: active(first(|0|(),X)) -> mark(nil())
r9: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r10: mark(terms(X)) -> active(terms(mark(X)))
r11: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r12: mark(recip(X)) -> active(recip(mark(X)))
r13: mark(sqr(X)) -> active(sqr(mark(X)))
r14: mark(s(X)) -> active(s(X))
r15: mark(|0|()) -> active(|0|())
r16: mark(add(X1,X2)) -> active(add(mark(X1),mark(X2)))
r17: mark(dbl(X)) -> active(dbl(mark(X)))
r18: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r19: mark(nil()) -> active(nil())
r20: terms(mark(X)) -> terms(X)
r21: terms(active(X)) -> terms(X)
r22: cons(mark(X1),X2) -> cons(X1,X2)
r23: cons(X1,mark(X2)) -> cons(X1,X2)
r24: cons(active(X1),X2) -> cons(X1,X2)
r25: cons(X1,active(X2)) -> cons(X1,X2)
r26: recip(mark(X)) -> recip(X)
r27: recip(active(X)) -> recip(X)
r28: sqr(mark(X)) -> sqr(X)
r29: sqr(active(X)) -> sqr(X)
r30: s(mark(X)) -> s(X)
r31: s(active(X)) -> s(X)
r32: add(mark(X1),X2) -> add(X1,X2)
r33: add(X1,mark(X2)) -> add(X1,X2)
r34: add(active(X1),X2) -> add(X1,X2)
r35: add(X1,active(X2)) -> add(X1,X2)
r36: dbl(mark(X)) -> dbl(X)
r37: dbl(active(X)) -> dbl(X)
r38: first(mark(X1),X2) -> first(X1,X2)
r39: first(X1,mark(X2)) -> first(X1,X2)
r40: first(active(X1),X2) -> first(X1,X2)
r41: first(X1,active(X2)) -> first(X1,X2)

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: recip#(active(X)) -> recip#(X)

and R consists of:

r1: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
r2: active(sqr(|0|())) -> mark(|0|())
r3: active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
r4: active(dbl(|0|())) -> mark(|0|())
r5: active(dbl(s(X))) -> mark(s(s(dbl(X))))
r6: active(add(|0|(),X)) -> mark(X)
r7: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r8: active(first(|0|(),X)) -> mark(nil())
r9: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r10: mark(terms(X)) -> active(terms(mark(X)))
r11: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r12: mark(recip(X)) -> active(recip(mark(X)))
r13: mark(sqr(X)) -> active(sqr(mark(X)))
r14: mark(s(X)) -> active(s(X))
r15: mark(|0|()) -> active(|0|())
r16: mark(add(X1,X2)) -> active(add(mark(X1),mark(X2)))
r17: mark(dbl(X)) -> active(dbl(mark(X)))
r18: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r19: mark(nil()) -> active(nil())
r20: terms(mark(X)) -> terms(X)
r21: terms(active(X)) -> terms(X)
r22: cons(mark(X1),X2) -> cons(X1,X2)
r23: cons(X1,mark(X2)) -> cons(X1,X2)
r24: cons(active(X1),X2) -> cons(X1,X2)
r25: cons(X1,active(X2)) -> cons(X1,X2)
r26: recip(mark(X)) -> recip(X)
r27: recip(active(X)) -> recip(X)
r28: sqr(mark(X)) -> sqr(X)
r29: sqr(active(X)) -> sqr(X)
r30: s(mark(X)) -> s(X)
r31: s(active(X)) -> s(X)
r32: add(mark(X1),X2) -> add(X1,X2)
r33: add(X1,mark(X2)) -> add(X1,X2)
r34: add(active(X1),X2) -> add(X1,X2)
r35: add(X1,active(X2)) -> add(X1,X2)
r36: dbl(mark(X)) -> dbl(X)
r37: dbl(active(X)) -> dbl(X)
r38: first(mark(X1),X2) -> first(X1,X2)
r39: first(X1,mark(X2)) -> first(X1,X2)
r40: first(active(X1),X2) -> first(X1,X2)
r41: first(X1,active(X2)) -> first(X1,X2)

The set of usable rules consists of

  (no rules)

Take the monotone reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        recip#_A(x1) = x1 + 1
        active_A(x1) = x1
  
    precedence:
    
      recip# = active
    
    partial status:
  
      pi(recip#) = [1]
      pi(active) = [1]

The next rules are strictly ordered:

  p1

We remove them from the problem.  Then no dependency pair remains.

-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: sqr#(mark(X)) -> sqr#(X)
p2: sqr#(active(X)) -> sqr#(X)

and R consists of:

r1: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
r2: active(sqr(|0|())) -> mark(|0|())
r3: active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
r4: active(dbl(|0|())) -> mark(|0|())
r5: active(dbl(s(X))) -> mark(s(s(dbl(X))))
r6: active(add(|0|(),X)) -> mark(X)
r7: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r8: active(first(|0|(),X)) -> mark(nil())
r9: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r10: mark(terms(X)) -> active(terms(mark(X)))
r11: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r12: mark(recip(X)) -> active(recip(mark(X)))
r13: mark(sqr(X)) -> active(sqr(mark(X)))
r14: mark(s(X)) -> active(s(X))
r15: mark(|0|()) -> active(|0|())
r16: mark(add(X1,X2)) -> active(add(mark(X1),mark(X2)))
r17: mark(dbl(X)) -> active(dbl(mark(X)))
r18: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r19: mark(nil()) -> active(nil())
r20: terms(mark(X)) -> terms(X)
r21: terms(active(X)) -> terms(X)
r22: cons(mark(X1),X2) -> cons(X1,X2)
r23: cons(X1,mark(X2)) -> cons(X1,X2)
r24: cons(active(X1),X2) -> cons(X1,X2)
r25: cons(X1,active(X2)) -> cons(X1,X2)
r26: recip(mark(X)) -> recip(X)
r27: recip(active(X)) -> recip(X)
r28: sqr(mark(X)) -> sqr(X)
r29: sqr(active(X)) -> sqr(X)
r30: s(mark(X)) -> s(X)
r31: s(active(X)) -> s(X)
r32: add(mark(X1),X2) -> add(X1,X2)
r33: add(X1,mark(X2)) -> add(X1,X2)
r34: add(active(X1),X2) -> add(X1,X2)
r35: add(X1,active(X2)) -> add(X1,X2)
r36: dbl(mark(X)) -> dbl(X)
r37: dbl(active(X)) -> dbl(X)
r38: first(mark(X1),X2) -> first(X1,X2)
r39: first(X1,mark(X2)) -> first(X1,X2)
r40: first(active(X1),X2) -> first(X1,X2)
r41: first(X1,active(X2)) -> first(X1,X2)

The set of usable rules consists of

  (no rules)

Take the monotone reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        sqr#_A(x1) = max{3, x1 + 2}
        mark_A(x1) = x1
        active_A(x1) = x1 + 1
  
    precedence:
    
      sqr# = mark = active
    
    partial status:
  
      pi(sqr#) = [1]
      pi(mark) = [1]
      pi(active) = [1]

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: sqr#(active(X)) -> sqr#(X)

and R consists of:

r1: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
r2: active(sqr(|0|())) -> mark(|0|())
r3: active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
r4: active(dbl(|0|())) -> mark(|0|())
r5: active(dbl(s(X))) -> mark(s(s(dbl(X))))
r6: active(add(|0|(),X)) -> mark(X)
r7: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r8: active(first(|0|(),X)) -> mark(nil())
r9: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r10: mark(terms(X)) -> active(terms(mark(X)))
r11: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r12: mark(recip(X)) -> active(recip(mark(X)))
r13: mark(sqr(X)) -> active(sqr(mark(X)))
r14: mark(s(X)) -> active(s(X))
r15: mark(|0|()) -> active(|0|())
r16: mark(add(X1,X2)) -> active(add(mark(X1),mark(X2)))
r17: mark(dbl(X)) -> active(dbl(mark(X)))
r18: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r19: mark(nil()) -> active(nil())
r20: terms(mark(X)) -> terms(X)
r21: terms(active(X)) -> terms(X)
r22: cons(mark(X1),X2) -> cons(X1,X2)
r23: cons(X1,mark(X2)) -> cons(X1,X2)
r24: cons(active(X1),X2) -> cons(X1,X2)
r25: cons(X1,active(X2)) -> cons(X1,X2)
r26: recip(mark(X)) -> recip(X)
r27: recip(active(X)) -> recip(X)
r28: sqr(mark(X)) -> sqr(X)
r29: sqr(active(X)) -> sqr(X)
r30: s(mark(X)) -> s(X)
r31: s(active(X)) -> s(X)
r32: add(mark(X1),X2) -> add(X1,X2)
r33: add(X1,mark(X2)) -> add(X1,X2)
r34: add(active(X1),X2) -> add(X1,X2)
r35: add(X1,active(X2)) -> add(X1,X2)
r36: dbl(mark(X)) -> dbl(X)
r37: dbl(active(X)) -> dbl(X)
r38: first(mark(X1),X2) -> first(X1,X2)
r39: first(X1,mark(X2)) -> first(X1,X2)
r40: first(active(X1),X2) -> first(X1,X2)
r41: first(X1,active(X2)) -> first(X1,X2)

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: sqr#(active(X)) -> sqr#(X)

and R consists of:

r1: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
r2: active(sqr(|0|())) -> mark(|0|())
r3: active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
r4: active(dbl(|0|())) -> mark(|0|())
r5: active(dbl(s(X))) -> mark(s(s(dbl(X))))
r6: active(add(|0|(),X)) -> mark(X)
r7: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r8: active(first(|0|(),X)) -> mark(nil())
r9: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r10: mark(terms(X)) -> active(terms(mark(X)))
r11: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r12: mark(recip(X)) -> active(recip(mark(X)))
r13: mark(sqr(X)) -> active(sqr(mark(X)))
r14: mark(s(X)) -> active(s(X))
r15: mark(|0|()) -> active(|0|())
r16: mark(add(X1,X2)) -> active(add(mark(X1),mark(X2)))
r17: mark(dbl(X)) -> active(dbl(mark(X)))
r18: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r19: mark(nil()) -> active(nil())
r20: terms(mark(X)) -> terms(X)
r21: terms(active(X)) -> terms(X)
r22: cons(mark(X1),X2) -> cons(X1,X2)
r23: cons(X1,mark(X2)) -> cons(X1,X2)
r24: cons(active(X1),X2) -> cons(X1,X2)
r25: cons(X1,active(X2)) -> cons(X1,X2)
r26: recip(mark(X)) -> recip(X)
r27: recip(active(X)) -> recip(X)
r28: sqr(mark(X)) -> sqr(X)
r29: sqr(active(X)) -> sqr(X)
r30: s(mark(X)) -> s(X)
r31: s(active(X)) -> s(X)
r32: add(mark(X1),X2) -> add(X1,X2)
r33: add(X1,mark(X2)) -> add(X1,X2)
r34: add(active(X1),X2) -> add(X1,X2)
r35: add(X1,active(X2)) -> add(X1,X2)
r36: dbl(mark(X)) -> dbl(X)
r37: dbl(active(X)) -> dbl(X)
r38: first(mark(X1),X2) -> first(X1,X2)
r39: first(X1,mark(X2)) -> first(X1,X2)
r40: first(active(X1),X2) -> first(X1,X2)
r41: first(X1,active(X2)) -> first(X1,X2)

The set of usable rules consists of

  (no rules)

Take the monotone reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        sqr#_A(x1) = x1 + 1
        active_A(x1) = x1
  
    precedence:
    
      sqr# = active
    
    partial status:
  
      pi(sqr#) = [1]
      pi(active) = [1]

The next rules are strictly ordered:

  p1

We remove them from the problem.  Then no dependency pair remains.

-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: terms#(mark(X)) -> terms#(X)
p2: terms#(active(X)) -> terms#(X)

and R consists of:

r1: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
r2: active(sqr(|0|())) -> mark(|0|())
r3: active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
r4: active(dbl(|0|())) -> mark(|0|())
r5: active(dbl(s(X))) -> mark(s(s(dbl(X))))
r6: active(add(|0|(),X)) -> mark(X)
r7: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r8: active(first(|0|(),X)) -> mark(nil())
r9: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r10: mark(terms(X)) -> active(terms(mark(X)))
r11: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r12: mark(recip(X)) -> active(recip(mark(X)))
r13: mark(sqr(X)) -> active(sqr(mark(X)))
r14: mark(s(X)) -> active(s(X))
r15: mark(|0|()) -> active(|0|())
r16: mark(add(X1,X2)) -> active(add(mark(X1),mark(X2)))
r17: mark(dbl(X)) -> active(dbl(mark(X)))
r18: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r19: mark(nil()) -> active(nil())
r20: terms(mark(X)) -> terms(X)
r21: terms(active(X)) -> terms(X)
r22: cons(mark(X1),X2) -> cons(X1,X2)
r23: cons(X1,mark(X2)) -> cons(X1,X2)
r24: cons(active(X1),X2) -> cons(X1,X2)
r25: cons(X1,active(X2)) -> cons(X1,X2)
r26: recip(mark(X)) -> recip(X)
r27: recip(active(X)) -> recip(X)
r28: sqr(mark(X)) -> sqr(X)
r29: sqr(active(X)) -> sqr(X)
r30: s(mark(X)) -> s(X)
r31: s(active(X)) -> s(X)
r32: add(mark(X1),X2) -> add(X1,X2)
r33: add(X1,mark(X2)) -> add(X1,X2)
r34: add(active(X1),X2) -> add(X1,X2)
r35: add(X1,active(X2)) -> add(X1,X2)
r36: dbl(mark(X)) -> dbl(X)
r37: dbl(active(X)) -> dbl(X)
r38: first(mark(X1),X2) -> first(X1,X2)
r39: first(X1,mark(X2)) -> first(X1,X2)
r40: first(active(X1),X2) -> first(X1,X2)
r41: first(X1,active(X2)) -> first(X1,X2)

The set of usable rules consists of

  (no rules)

Take the monotone reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        terms#_A(x1) = max{3, x1 + 2}
        mark_A(x1) = x1
        active_A(x1) = x1 + 1
  
    precedence:
    
      terms# = mark = active
    
    partial status:
  
      pi(terms#) = [1]
      pi(mark) = [1]
      pi(active) = [1]

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: terms#(active(X)) -> terms#(X)

and R consists of:

r1: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
r2: active(sqr(|0|())) -> mark(|0|())
r3: active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
r4: active(dbl(|0|())) -> mark(|0|())
r5: active(dbl(s(X))) -> mark(s(s(dbl(X))))
r6: active(add(|0|(),X)) -> mark(X)
r7: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r8: active(first(|0|(),X)) -> mark(nil())
r9: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r10: mark(terms(X)) -> active(terms(mark(X)))
r11: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r12: mark(recip(X)) -> active(recip(mark(X)))
r13: mark(sqr(X)) -> active(sqr(mark(X)))
r14: mark(s(X)) -> active(s(X))
r15: mark(|0|()) -> active(|0|())
r16: mark(add(X1,X2)) -> active(add(mark(X1),mark(X2)))
r17: mark(dbl(X)) -> active(dbl(mark(X)))
r18: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r19: mark(nil()) -> active(nil())
r20: terms(mark(X)) -> terms(X)
r21: terms(active(X)) -> terms(X)
r22: cons(mark(X1),X2) -> cons(X1,X2)
r23: cons(X1,mark(X2)) -> cons(X1,X2)
r24: cons(active(X1),X2) -> cons(X1,X2)
r25: cons(X1,active(X2)) -> cons(X1,X2)
r26: recip(mark(X)) -> recip(X)
r27: recip(active(X)) -> recip(X)
r28: sqr(mark(X)) -> sqr(X)
r29: sqr(active(X)) -> sqr(X)
r30: s(mark(X)) -> s(X)
r31: s(active(X)) -> s(X)
r32: add(mark(X1),X2) -> add(X1,X2)
r33: add(X1,mark(X2)) -> add(X1,X2)
r34: add(active(X1),X2) -> add(X1,X2)
r35: add(X1,active(X2)) -> add(X1,X2)
r36: dbl(mark(X)) -> dbl(X)
r37: dbl(active(X)) -> dbl(X)
r38: first(mark(X1),X2) -> first(X1,X2)
r39: first(X1,mark(X2)) -> first(X1,X2)
r40: first(active(X1),X2) -> first(X1,X2)
r41: first(X1,active(X2)) -> first(X1,X2)

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: terms#(active(X)) -> terms#(X)

and R consists of:

r1: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
r2: active(sqr(|0|())) -> mark(|0|())
r3: active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
r4: active(dbl(|0|())) -> mark(|0|())
r5: active(dbl(s(X))) -> mark(s(s(dbl(X))))
r6: active(add(|0|(),X)) -> mark(X)
r7: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r8: active(first(|0|(),X)) -> mark(nil())
r9: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r10: mark(terms(X)) -> active(terms(mark(X)))
r11: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r12: mark(recip(X)) -> active(recip(mark(X)))
r13: mark(sqr(X)) -> active(sqr(mark(X)))
r14: mark(s(X)) -> active(s(X))
r15: mark(|0|()) -> active(|0|())
r16: mark(add(X1,X2)) -> active(add(mark(X1),mark(X2)))
r17: mark(dbl(X)) -> active(dbl(mark(X)))
r18: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r19: mark(nil()) -> active(nil())
r20: terms(mark(X)) -> terms(X)
r21: terms(active(X)) -> terms(X)
r22: cons(mark(X1),X2) -> cons(X1,X2)
r23: cons(X1,mark(X2)) -> cons(X1,X2)
r24: cons(active(X1),X2) -> cons(X1,X2)
r25: cons(X1,active(X2)) -> cons(X1,X2)
r26: recip(mark(X)) -> recip(X)
r27: recip(active(X)) -> recip(X)
r28: sqr(mark(X)) -> sqr(X)
r29: sqr(active(X)) -> sqr(X)
r30: s(mark(X)) -> s(X)
r31: s(active(X)) -> s(X)
r32: add(mark(X1),X2) -> add(X1,X2)
r33: add(X1,mark(X2)) -> add(X1,X2)
r34: add(active(X1),X2) -> add(X1,X2)
r35: add(X1,active(X2)) -> add(X1,X2)
r36: dbl(mark(X)) -> dbl(X)
r37: dbl(active(X)) -> dbl(X)
r38: first(mark(X1),X2) -> first(X1,X2)
r39: first(X1,mark(X2)) -> first(X1,X2)
r40: first(active(X1),X2) -> first(X1,X2)
r41: first(X1,active(X2)) -> first(X1,X2)

The set of usable rules consists of

  (no rules)

Take the monotone reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        terms#_A(x1) = x1 + 1
        active_A(x1) = x1
  
    precedence:
    
      terms# = active
    
    partial status:
  
      pi(terms#) = [1]
      pi(active) = [1]

The next rules are strictly ordered:

  p1

We remove them from the problem.  Then no dependency pair remains.

-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: s#(mark(X)) -> s#(X)
p2: s#(active(X)) -> s#(X)

and R consists of:

r1: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
r2: active(sqr(|0|())) -> mark(|0|())
r3: active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
r4: active(dbl(|0|())) -> mark(|0|())
r5: active(dbl(s(X))) -> mark(s(s(dbl(X))))
r6: active(add(|0|(),X)) -> mark(X)
r7: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r8: active(first(|0|(),X)) -> mark(nil())
r9: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r10: mark(terms(X)) -> active(terms(mark(X)))
r11: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r12: mark(recip(X)) -> active(recip(mark(X)))
r13: mark(sqr(X)) -> active(sqr(mark(X)))
r14: mark(s(X)) -> active(s(X))
r15: mark(|0|()) -> active(|0|())
r16: mark(add(X1,X2)) -> active(add(mark(X1),mark(X2)))
r17: mark(dbl(X)) -> active(dbl(mark(X)))
r18: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r19: mark(nil()) -> active(nil())
r20: terms(mark(X)) -> terms(X)
r21: terms(active(X)) -> terms(X)
r22: cons(mark(X1),X2) -> cons(X1,X2)
r23: cons(X1,mark(X2)) -> cons(X1,X2)
r24: cons(active(X1),X2) -> cons(X1,X2)
r25: cons(X1,active(X2)) -> cons(X1,X2)
r26: recip(mark(X)) -> recip(X)
r27: recip(active(X)) -> recip(X)
r28: sqr(mark(X)) -> sqr(X)
r29: sqr(active(X)) -> sqr(X)
r30: s(mark(X)) -> s(X)
r31: s(active(X)) -> s(X)
r32: add(mark(X1),X2) -> add(X1,X2)
r33: add(X1,mark(X2)) -> add(X1,X2)
r34: add(active(X1),X2) -> add(X1,X2)
r35: add(X1,active(X2)) -> add(X1,X2)
r36: dbl(mark(X)) -> dbl(X)
r37: dbl(active(X)) -> dbl(X)
r38: first(mark(X1),X2) -> first(X1,X2)
r39: first(X1,mark(X2)) -> first(X1,X2)
r40: first(active(X1),X2) -> first(X1,X2)
r41: first(X1,active(X2)) -> first(X1,X2)

The set of usable rules consists of

  (no rules)

Take the monotone reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        s#_A(x1) = max{3, x1 + 2}
        mark_A(x1) = x1
        active_A(x1) = x1 + 1
  
    precedence:
    
      s# = mark = active
    
    partial status:
  
      pi(s#) = [1]
      pi(mark) = [1]
      pi(active) = [1]

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: s#(active(X)) -> s#(X)

and R consists of:

r1: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
r2: active(sqr(|0|())) -> mark(|0|())
r3: active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
r4: active(dbl(|0|())) -> mark(|0|())
r5: active(dbl(s(X))) -> mark(s(s(dbl(X))))
r6: active(add(|0|(),X)) -> mark(X)
r7: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r8: active(first(|0|(),X)) -> mark(nil())
r9: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r10: mark(terms(X)) -> active(terms(mark(X)))
r11: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r12: mark(recip(X)) -> active(recip(mark(X)))
r13: mark(sqr(X)) -> active(sqr(mark(X)))
r14: mark(s(X)) -> active(s(X))
r15: mark(|0|()) -> active(|0|())
r16: mark(add(X1,X2)) -> active(add(mark(X1),mark(X2)))
r17: mark(dbl(X)) -> active(dbl(mark(X)))
r18: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r19: mark(nil()) -> active(nil())
r20: terms(mark(X)) -> terms(X)
r21: terms(active(X)) -> terms(X)
r22: cons(mark(X1),X2) -> cons(X1,X2)
r23: cons(X1,mark(X2)) -> cons(X1,X2)
r24: cons(active(X1),X2) -> cons(X1,X2)
r25: cons(X1,active(X2)) -> cons(X1,X2)
r26: recip(mark(X)) -> recip(X)
r27: recip(active(X)) -> recip(X)
r28: sqr(mark(X)) -> sqr(X)
r29: sqr(active(X)) -> sqr(X)
r30: s(mark(X)) -> s(X)
r31: s(active(X)) -> s(X)
r32: add(mark(X1),X2) -> add(X1,X2)
r33: add(X1,mark(X2)) -> add(X1,X2)
r34: add(active(X1),X2) -> add(X1,X2)
r35: add(X1,active(X2)) -> add(X1,X2)
r36: dbl(mark(X)) -> dbl(X)
r37: dbl(active(X)) -> dbl(X)
r38: first(mark(X1),X2) -> first(X1,X2)
r39: first(X1,mark(X2)) -> first(X1,X2)
r40: first(active(X1),X2) -> first(X1,X2)
r41: first(X1,active(X2)) -> first(X1,X2)

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: s#(active(X)) -> s#(X)

and R consists of:

r1: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
r2: active(sqr(|0|())) -> mark(|0|())
r3: active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
r4: active(dbl(|0|())) -> mark(|0|())
r5: active(dbl(s(X))) -> mark(s(s(dbl(X))))
r6: active(add(|0|(),X)) -> mark(X)
r7: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r8: active(first(|0|(),X)) -> mark(nil())
r9: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r10: mark(terms(X)) -> active(terms(mark(X)))
r11: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r12: mark(recip(X)) -> active(recip(mark(X)))
r13: mark(sqr(X)) -> active(sqr(mark(X)))
r14: mark(s(X)) -> active(s(X))
r15: mark(|0|()) -> active(|0|())
r16: mark(add(X1,X2)) -> active(add(mark(X1),mark(X2)))
r17: mark(dbl(X)) -> active(dbl(mark(X)))
r18: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r19: mark(nil()) -> active(nil())
r20: terms(mark(X)) -> terms(X)
r21: terms(active(X)) -> terms(X)
r22: cons(mark(X1),X2) -> cons(X1,X2)
r23: cons(X1,mark(X2)) -> cons(X1,X2)
r24: cons(active(X1),X2) -> cons(X1,X2)
r25: cons(X1,active(X2)) -> cons(X1,X2)
r26: recip(mark(X)) -> recip(X)
r27: recip(active(X)) -> recip(X)
r28: sqr(mark(X)) -> sqr(X)
r29: sqr(active(X)) -> sqr(X)
r30: s(mark(X)) -> s(X)
r31: s(active(X)) -> s(X)
r32: add(mark(X1),X2) -> add(X1,X2)
r33: add(X1,mark(X2)) -> add(X1,X2)
r34: add(active(X1),X2) -> add(X1,X2)
r35: add(X1,active(X2)) -> add(X1,X2)
r36: dbl(mark(X)) -> dbl(X)
r37: dbl(active(X)) -> dbl(X)
r38: first(mark(X1),X2) -> first(X1,X2)
r39: first(X1,mark(X2)) -> first(X1,X2)
r40: first(active(X1),X2) -> first(X1,X2)
r41: first(X1,active(X2)) -> first(X1,X2)

The set of usable rules consists of

  (no rules)

Take the monotone reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        s#_A(x1) = x1 + 1
        active_A(x1) = x1
  
    precedence:
    
      s# = active
    
    partial status:
  
      pi(s#) = [1]
      pi(active) = [1]

The next rules are strictly ordered:

  p1

We remove them from the problem.  Then no dependency pair remains.

-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: add#(mark(X1),X2) -> add#(X1,X2)
p2: add#(X1,active(X2)) -> add#(X1,X2)
p3: add#(active(X1),X2) -> add#(X1,X2)
p4: add#(X1,mark(X2)) -> add#(X1,X2)

and R consists of:

r1: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
r2: active(sqr(|0|())) -> mark(|0|())
r3: active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
r4: active(dbl(|0|())) -> mark(|0|())
r5: active(dbl(s(X))) -> mark(s(s(dbl(X))))
r6: active(add(|0|(),X)) -> mark(X)
r7: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r8: active(first(|0|(),X)) -> mark(nil())
r9: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r10: mark(terms(X)) -> active(terms(mark(X)))
r11: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r12: mark(recip(X)) -> active(recip(mark(X)))
r13: mark(sqr(X)) -> active(sqr(mark(X)))
r14: mark(s(X)) -> active(s(X))
r15: mark(|0|()) -> active(|0|())
r16: mark(add(X1,X2)) -> active(add(mark(X1),mark(X2)))
r17: mark(dbl(X)) -> active(dbl(mark(X)))
r18: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r19: mark(nil()) -> active(nil())
r20: terms(mark(X)) -> terms(X)
r21: terms(active(X)) -> terms(X)
r22: cons(mark(X1),X2) -> cons(X1,X2)
r23: cons(X1,mark(X2)) -> cons(X1,X2)
r24: cons(active(X1),X2) -> cons(X1,X2)
r25: cons(X1,active(X2)) -> cons(X1,X2)
r26: recip(mark(X)) -> recip(X)
r27: recip(active(X)) -> recip(X)
r28: sqr(mark(X)) -> sqr(X)
r29: sqr(active(X)) -> sqr(X)
r30: s(mark(X)) -> s(X)
r31: s(active(X)) -> s(X)
r32: add(mark(X1),X2) -> add(X1,X2)
r33: add(X1,mark(X2)) -> add(X1,X2)
r34: add(active(X1),X2) -> add(X1,X2)
r35: add(X1,active(X2)) -> add(X1,X2)
r36: dbl(mark(X)) -> dbl(X)
r37: dbl(active(X)) -> dbl(X)
r38: first(mark(X1),X2) -> first(X1,X2)
r39: first(X1,mark(X2)) -> first(X1,X2)
r40: first(active(X1),X2) -> first(X1,X2)
r41: first(X1,active(X2)) -> first(X1,X2)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        add#_A(x1,x2) = max{0, x2 - 2}
        mark_A(x1) = x1
        active_A(x1) = max{3, x1 + 1}
  
    precedence:
    
      add# = mark = active
    
    partial status:
  
      pi(add#) = []
      pi(mark) = [1]
      pi(active) = [1]

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: add#(mark(X1),X2) -> add#(X1,X2)
p2: add#(active(X1),X2) -> add#(X1,X2)
p3: add#(X1,mark(X2)) -> add#(X1,X2)

and R consists of:

r1: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
r2: active(sqr(|0|())) -> mark(|0|())
r3: active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
r4: active(dbl(|0|())) -> mark(|0|())
r5: active(dbl(s(X))) -> mark(s(s(dbl(X))))
r6: active(add(|0|(),X)) -> mark(X)
r7: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r8: active(first(|0|(),X)) -> mark(nil())
r9: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r10: mark(terms(X)) -> active(terms(mark(X)))
r11: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r12: mark(recip(X)) -> active(recip(mark(X)))
r13: mark(sqr(X)) -> active(sqr(mark(X)))
r14: mark(s(X)) -> active(s(X))
r15: mark(|0|()) -> active(|0|())
r16: mark(add(X1,X2)) -> active(add(mark(X1),mark(X2)))
r17: mark(dbl(X)) -> active(dbl(mark(X)))
r18: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r19: mark(nil()) -> active(nil())
r20: terms(mark(X)) -> terms(X)
r21: terms(active(X)) -> terms(X)
r22: cons(mark(X1),X2) -> cons(X1,X2)
r23: cons(X1,mark(X2)) -> cons(X1,X2)
r24: cons(active(X1),X2) -> cons(X1,X2)
r25: cons(X1,active(X2)) -> cons(X1,X2)
r26: recip(mark(X)) -> recip(X)
r27: recip(active(X)) -> recip(X)
r28: sqr(mark(X)) -> sqr(X)
r29: sqr(active(X)) -> sqr(X)
r30: s(mark(X)) -> s(X)
r31: s(active(X)) -> s(X)
r32: add(mark(X1),X2) -> add(X1,X2)
r33: add(X1,mark(X2)) -> add(X1,X2)
r34: add(active(X1),X2) -> add(X1,X2)
r35: add(X1,active(X2)) -> add(X1,X2)
r36: dbl(mark(X)) -> dbl(X)
r37: dbl(active(X)) -> dbl(X)
r38: first(mark(X1),X2) -> first(X1,X2)
r39: first(X1,mark(X2)) -> first(X1,X2)
r40: first(active(X1),X2) -> first(X1,X2)
r41: first(X1,active(X2)) -> first(X1,X2)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: add#(mark(X1),X2) -> add#(X1,X2)
p2: add#(X1,mark(X2)) -> add#(X1,X2)
p3: add#(active(X1),X2) -> add#(X1,X2)

and R consists of:

r1: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
r2: active(sqr(|0|())) -> mark(|0|())
r3: active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
r4: active(dbl(|0|())) -> mark(|0|())
r5: active(dbl(s(X))) -> mark(s(s(dbl(X))))
r6: active(add(|0|(),X)) -> mark(X)
r7: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r8: active(first(|0|(),X)) -> mark(nil())
r9: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r10: mark(terms(X)) -> active(terms(mark(X)))
r11: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r12: mark(recip(X)) -> active(recip(mark(X)))
r13: mark(sqr(X)) -> active(sqr(mark(X)))
r14: mark(s(X)) -> active(s(X))
r15: mark(|0|()) -> active(|0|())
r16: mark(add(X1,X2)) -> active(add(mark(X1),mark(X2)))
r17: mark(dbl(X)) -> active(dbl(mark(X)))
r18: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r19: mark(nil()) -> active(nil())
r20: terms(mark(X)) -> terms(X)
r21: terms(active(X)) -> terms(X)
r22: cons(mark(X1),X2) -> cons(X1,X2)
r23: cons(X1,mark(X2)) -> cons(X1,X2)
r24: cons(active(X1),X2) -> cons(X1,X2)
r25: cons(X1,active(X2)) -> cons(X1,X2)
r26: recip(mark(X)) -> recip(X)
r27: recip(active(X)) -> recip(X)
r28: sqr(mark(X)) -> sqr(X)
r29: sqr(active(X)) -> sqr(X)
r30: s(mark(X)) -> s(X)
r31: s(active(X)) -> s(X)
r32: add(mark(X1),X2) -> add(X1,X2)
r33: add(X1,mark(X2)) -> add(X1,X2)
r34: add(active(X1),X2) -> add(X1,X2)
r35: add(X1,active(X2)) -> add(X1,X2)
r36: dbl(mark(X)) -> dbl(X)
r37: dbl(active(X)) -> dbl(X)
r38: first(mark(X1),X2) -> first(X1,X2)
r39: first(X1,mark(X2)) -> first(X1,X2)
r40: first(active(X1),X2) -> first(X1,X2)
r41: first(X1,active(X2)) -> first(X1,X2)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        add#_A(x1,x2) = max{0, x1 - 2}
        mark_A(x1) = max{3, x1 + 1}
        active_A(x1) = x1
  
    precedence:
    
      add# = mark = active
    
    partial status:
  
      pi(add#) = []
      pi(mark) = [1]
      pi(active) = [1]

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: add#(X1,mark(X2)) -> add#(X1,X2)
p2: add#(active(X1),X2) -> add#(X1,X2)

and R consists of:

r1: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
r2: active(sqr(|0|())) -> mark(|0|())
r3: active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
r4: active(dbl(|0|())) -> mark(|0|())
r5: active(dbl(s(X))) -> mark(s(s(dbl(X))))
r6: active(add(|0|(),X)) -> mark(X)
r7: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r8: active(first(|0|(),X)) -> mark(nil())
r9: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r10: mark(terms(X)) -> active(terms(mark(X)))
r11: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r12: mark(recip(X)) -> active(recip(mark(X)))
r13: mark(sqr(X)) -> active(sqr(mark(X)))
r14: mark(s(X)) -> active(s(X))
r15: mark(|0|()) -> active(|0|())
r16: mark(add(X1,X2)) -> active(add(mark(X1),mark(X2)))
r17: mark(dbl(X)) -> active(dbl(mark(X)))
r18: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r19: mark(nil()) -> active(nil())
r20: terms(mark(X)) -> terms(X)
r21: terms(active(X)) -> terms(X)
r22: cons(mark(X1),X2) -> cons(X1,X2)
r23: cons(X1,mark(X2)) -> cons(X1,X2)
r24: cons(active(X1),X2) -> cons(X1,X2)
r25: cons(X1,active(X2)) -> cons(X1,X2)
r26: recip(mark(X)) -> recip(X)
r27: recip(active(X)) -> recip(X)
r28: sqr(mark(X)) -> sqr(X)
r29: sqr(active(X)) -> sqr(X)
r30: s(mark(X)) -> s(X)
r31: s(active(X)) -> s(X)
r32: add(mark(X1),X2) -> add(X1,X2)
r33: add(X1,mark(X2)) -> add(X1,X2)
r34: add(active(X1),X2) -> add(X1,X2)
r35: add(X1,active(X2)) -> add(X1,X2)
r36: dbl(mark(X)) -> dbl(X)
r37: dbl(active(X)) -> dbl(X)
r38: first(mark(X1),X2) -> first(X1,X2)
r39: first(X1,mark(X2)) -> first(X1,X2)
r40: first(active(X1),X2) -> first(X1,X2)
r41: first(X1,active(X2)) -> first(X1,X2)

The estimated dependency graph contains the following SCCs:

  {p1, p2}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: add#(X1,mark(X2)) -> add#(X1,X2)
p2: add#(active(X1),X2) -> add#(X1,X2)

and R consists of:

r1: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
r2: active(sqr(|0|())) -> mark(|0|())
r3: active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
r4: active(dbl(|0|())) -> mark(|0|())
r5: active(dbl(s(X))) -> mark(s(s(dbl(X))))
r6: active(add(|0|(),X)) -> mark(X)
r7: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r8: active(first(|0|(),X)) -> mark(nil())
r9: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r10: mark(terms(X)) -> active(terms(mark(X)))
r11: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r12: mark(recip(X)) -> active(recip(mark(X)))
r13: mark(sqr(X)) -> active(sqr(mark(X)))
r14: mark(s(X)) -> active(s(X))
r15: mark(|0|()) -> active(|0|())
r16: mark(add(X1,X2)) -> active(add(mark(X1),mark(X2)))
r17: mark(dbl(X)) -> active(dbl(mark(X)))
r18: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r19: mark(nil()) -> active(nil())
r20: terms(mark(X)) -> terms(X)
r21: terms(active(X)) -> terms(X)
r22: cons(mark(X1),X2) -> cons(X1,X2)
r23: cons(X1,mark(X2)) -> cons(X1,X2)
r24: cons(active(X1),X2) -> cons(X1,X2)
r25: cons(X1,active(X2)) -> cons(X1,X2)
r26: recip(mark(X)) -> recip(X)
r27: recip(active(X)) -> recip(X)
r28: sqr(mark(X)) -> sqr(X)
r29: sqr(active(X)) -> sqr(X)
r30: s(mark(X)) -> s(X)
r31: s(active(X)) -> s(X)
r32: add(mark(X1),X2) -> add(X1,X2)
r33: add(X1,mark(X2)) -> add(X1,X2)
r34: add(active(X1),X2) -> add(X1,X2)
r35: add(X1,active(X2)) -> add(X1,X2)
r36: dbl(mark(X)) -> dbl(X)
r37: dbl(active(X)) -> dbl(X)
r38: first(mark(X1),X2) -> first(X1,X2)
r39: first(X1,mark(X2)) -> first(X1,X2)
r40: first(active(X1),X2) -> first(X1,X2)
r41: first(X1,active(X2)) -> first(X1,X2)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        add#_A(x1,x2) = x2 + 1
        mark_A(x1) = x1 + 1
        active_A(x1) = x1 + 1
  
    precedence:
    
      add# = mark = active
    
    partial status:
  
      pi(add#) = []
      pi(mark) = [1]
      pi(active) = [1]

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: add#(active(X1),X2) -> add#(X1,X2)

and R consists of:

r1: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
r2: active(sqr(|0|())) -> mark(|0|())
r3: active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
r4: active(dbl(|0|())) -> mark(|0|())
r5: active(dbl(s(X))) -> mark(s(s(dbl(X))))
r6: active(add(|0|(),X)) -> mark(X)
r7: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r8: active(first(|0|(),X)) -> mark(nil())
r9: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r10: mark(terms(X)) -> active(terms(mark(X)))
r11: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r12: mark(recip(X)) -> active(recip(mark(X)))
r13: mark(sqr(X)) -> active(sqr(mark(X)))
r14: mark(s(X)) -> active(s(X))
r15: mark(|0|()) -> active(|0|())
r16: mark(add(X1,X2)) -> active(add(mark(X1),mark(X2)))
r17: mark(dbl(X)) -> active(dbl(mark(X)))
r18: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r19: mark(nil()) -> active(nil())
r20: terms(mark(X)) -> terms(X)
r21: terms(active(X)) -> terms(X)
r22: cons(mark(X1),X2) -> cons(X1,X2)
r23: cons(X1,mark(X2)) -> cons(X1,X2)
r24: cons(active(X1),X2) -> cons(X1,X2)
r25: cons(X1,active(X2)) -> cons(X1,X2)
r26: recip(mark(X)) -> recip(X)
r27: recip(active(X)) -> recip(X)
r28: sqr(mark(X)) -> sqr(X)
r29: sqr(active(X)) -> sqr(X)
r30: s(mark(X)) -> s(X)
r31: s(active(X)) -> s(X)
r32: add(mark(X1),X2) -> add(X1,X2)
r33: add(X1,mark(X2)) -> add(X1,X2)
r34: add(active(X1),X2) -> add(X1,X2)
r35: add(X1,active(X2)) -> add(X1,X2)
r36: dbl(mark(X)) -> dbl(X)
r37: dbl(active(X)) -> dbl(X)
r38: first(mark(X1),X2) -> first(X1,X2)
r39: first(X1,mark(X2)) -> first(X1,X2)
r40: first(active(X1),X2) -> first(X1,X2)
r41: first(X1,active(X2)) -> first(X1,X2)

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: add#(active(X1),X2) -> add#(X1,X2)

and R consists of:

r1: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
r2: active(sqr(|0|())) -> mark(|0|())
r3: active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
r4: active(dbl(|0|())) -> mark(|0|())
r5: active(dbl(s(X))) -> mark(s(s(dbl(X))))
r6: active(add(|0|(),X)) -> mark(X)
r7: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r8: active(first(|0|(),X)) -> mark(nil())
r9: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r10: mark(terms(X)) -> active(terms(mark(X)))
r11: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r12: mark(recip(X)) -> active(recip(mark(X)))
r13: mark(sqr(X)) -> active(sqr(mark(X)))
r14: mark(s(X)) -> active(s(X))
r15: mark(|0|()) -> active(|0|())
r16: mark(add(X1,X2)) -> active(add(mark(X1),mark(X2)))
r17: mark(dbl(X)) -> active(dbl(mark(X)))
r18: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r19: mark(nil()) -> active(nil())
r20: terms(mark(X)) -> terms(X)
r21: terms(active(X)) -> terms(X)
r22: cons(mark(X1),X2) -> cons(X1,X2)
r23: cons(X1,mark(X2)) -> cons(X1,X2)
r24: cons(active(X1),X2) -> cons(X1,X2)
r25: cons(X1,active(X2)) -> cons(X1,X2)
r26: recip(mark(X)) -> recip(X)
r27: recip(active(X)) -> recip(X)
r28: sqr(mark(X)) -> sqr(X)
r29: sqr(active(X)) -> sqr(X)
r30: s(mark(X)) -> s(X)
r31: s(active(X)) -> s(X)
r32: add(mark(X1),X2) -> add(X1,X2)
r33: add(X1,mark(X2)) -> add(X1,X2)
r34: add(active(X1),X2) -> add(X1,X2)
r35: add(X1,active(X2)) -> add(X1,X2)
r36: dbl(mark(X)) -> dbl(X)
r37: dbl(active(X)) -> dbl(X)
r38: first(mark(X1),X2) -> first(X1,X2)
r39: first(X1,mark(X2)) -> first(X1,X2)
r40: first(active(X1),X2) -> first(X1,X2)
r41: first(X1,active(X2)) -> first(X1,X2)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        add#_A(x1,x2) = max{1, x1, x2}
        active_A(x1) = x1 + 1
  
    precedence:
    
      add# = active
    
    partial status:
  
      pi(add#) = [1, 2]
      pi(active) = []

The next rules are strictly ordered:

  p1

We remove them from the problem.  Then no dependency pair remains.

-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: dbl#(mark(X)) -> dbl#(X)
p2: dbl#(active(X)) -> dbl#(X)

and R consists of:

r1: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
r2: active(sqr(|0|())) -> mark(|0|())
r3: active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
r4: active(dbl(|0|())) -> mark(|0|())
r5: active(dbl(s(X))) -> mark(s(s(dbl(X))))
r6: active(add(|0|(),X)) -> mark(X)
r7: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r8: active(first(|0|(),X)) -> mark(nil())
r9: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r10: mark(terms(X)) -> active(terms(mark(X)))
r11: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r12: mark(recip(X)) -> active(recip(mark(X)))
r13: mark(sqr(X)) -> active(sqr(mark(X)))
r14: mark(s(X)) -> active(s(X))
r15: mark(|0|()) -> active(|0|())
r16: mark(add(X1,X2)) -> active(add(mark(X1),mark(X2)))
r17: mark(dbl(X)) -> active(dbl(mark(X)))
r18: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r19: mark(nil()) -> active(nil())
r20: terms(mark(X)) -> terms(X)
r21: terms(active(X)) -> terms(X)
r22: cons(mark(X1),X2) -> cons(X1,X2)
r23: cons(X1,mark(X2)) -> cons(X1,X2)
r24: cons(active(X1),X2) -> cons(X1,X2)
r25: cons(X1,active(X2)) -> cons(X1,X2)
r26: recip(mark(X)) -> recip(X)
r27: recip(active(X)) -> recip(X)
r28: sqr(mark(X)) -> sqr(X)
r29: sqr(active(X)) -> sqr(X)
r30: s(mark(X)) -> s(X)
r31: s(active(X)) -> s(X)
r32: add(mark(X1),X2) -> add(X1,X2)
r33: add(X1,mark(X2)) -> add(X1,X2)
r34: add(active(X1),X2) -> add(X1,X2)
r35: add(X1,active(X2)) -> add(X1,X2)
r36: dbl(mark(X)) -> dbl(X)
r37: dbl(active(X)) -> dbl(X)
r38: first(mark(X1),X2) -> first(X1,X2)
r39: first(X1,mark(X2)) -> first(X1,X2)
r40: first(active(X1),X2) -> first(X1,X2)
r41: first(X1,active(X2)) -> first(X1,X2)

The set of usable rules consists of

  (no rules)

Take the monotone reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        dbl#_A(x1) = x1 + 1
        mark_A(x1) = x1
        active_A(x1) = x1
  
    precedence:
    
      dbl# = mark = active
    
    partial status:
  
      pi(dbl#) = [1]
      pi(mark) = [1]
      pi(active) = [1]

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: dbl#(active(X)) -> dbl#(X)

and R consists of:

r1: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
r2: active(sqr(|0|())) -> mark(|0|())
r3: active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
r4: active(dbl(|0|())) -> mark(|0|())
r5: active(dbl(s(X))) -> mark(s(s(dbl(X))))
r6: active(add(|0|(),X)) -> mark(X)
r7: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r8: active(first(|0|(),X)) -> mark(nil())
r9: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r10: mark(terms(X)) -> active(terms(mark(X)))
r11: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r12: mark(recip(X)) -> active(recip(mark(X)))
r13: mark(sqr(X)) -> active(sqr(mark(X)))
r14: mark(s(X)) -> active(s(X))
r15: mark(|0|()) -> active(|0|())
r16: mark(add(X1,X2)) -> active(add(mark(X1),mark(X2)))
r17: mark(dbl(X)) -> active(dbl(mark(X)))
r18: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r19: mark(nil()) -> active(nil())
r20: terms(mark(X)) -> terms(X)
r21: terms(active(X)) -> terms(X)
r22: cons(mark(X1),X2) -> cons(X1,X2)
r23: cons(X1,mark(X2)) -> cons(X1,X2)
r24: cons(active(X1),X2) -> cons(X1,X2)
r25: cons(X1,active(X2)) -> cons(X1,X2)
r26: recip(mark(X)) -> recip(X)
r27: recip(active(X)) -> recip(X)
r28: sqr(mark(X)) -> sqr(X)
r29: sqr(active(X)) -> sqr(X)
r30: s(mark(X)) -> s(X)
r31: s(active(X)) -> s(X)
r32: add(mark(X1),X2) -> add(X1,X2)
r33: add(X1,mark(X2)) -> add(X1,X2)
r34: add(active(X1),X2) -> add(X1,X2)
r35: add(X1,active(X2)) -> add(X1,X2)
r36: dbl(mark(X)) -> dbl(X)
r37: dbl(active(X)) -> dbl(X)
r38: first(mark(X1),X2) -> first(X1,X2)
r39: first(X1,mark(X2)) -> first(X1,X2)
r40: first(active(X1),X2) -> first(X1,X2)
r41: first(X1,active(X2)) -> first(X1,X2)

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: dbl#(active(X)) -> dbl#(X)

and R consists of:

r1: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
r2: active(sqr(|0|())) -> mark(|0|())
r3: active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
r4: active(dbl(|0|())) -> mark(|0|())
r5: active(dbl(s(X))) -> mark(s(s(dbl(X))))
r6: active(add(|0|(),X)) -> mark(X)
r7: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r8: active(first(|0|(),X)) -> mark(nil())
r9: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r10: mark(terms(X)) -> active(terms(mark(X)))
r11: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r12: mark(recip(X)) -> active(recip(mark(X)))
r13: mark(sqr(X)) -> active(sqr(mark(X)))
r14: mark(s(X)) -> active(s(X))
r15: mark(|0|()) -> active(|0|())
r16: mark(add(X1,X2)) -> active(add(mark(X1),mark(X2)))
r17: mark(dbl(X)) -> active(dbl(mark(X)))
r18: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r19: mark(nil()) -> active(nil())
r20: terms(mark(X)) -> terms(X)
r21: terms(active(X)) -> terms(X)
r22: cons(mark(X1),X2) -> cons(X1,X2)
r23: cons(X1,mark(X2)) -> cons(X1,X2)
r24: cons(active(X1),X2) -> cons(X1,X2)
r25: cons(X1,active(X2)) -> cons(X1,X2)
r26: recip(mark(X)) -> recip(X)
r27: recip(active(X)) -> recip(X)
r28: sqr(mark(X)) -> sqr(X)
r29: sqr(active(X)) -> sqr(X)
r30: s(mark(X)) -> s(X)
r31: s(active(X)) -> s(X)
r32: add(mark(X1),X2) -> add(X1,X2)
r33: add(X1,mark(X2)) -> add(X1,X2)
r34: add(active(X1),X2) -> add(X1,X2)
r35: add(X1,active(X2)) -> add(X1,X2)
r36: dbl(mark(X)) -> dbl(X)
r37: dbl(active(X)) -> dbl(X)
r38: first(mark(X1),X2) -> first(X1,X2)
r39: first(X1,mark(X2)) -> first(X1,X2)
r40: first(active(X1),X2) -> first(X1,X2)
r41: first(X1,active(X2)) -> first(X1,X2)

The set of usable rules consists of

  (no rules)

Take the monotone reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        dbl#_A(x1) = x1 + 1
        active_A(x1) = x1
  
    precedence:
    
      dbl# = active
    
    partial status:
  
      pi(dbl#) = [1]
      pi(active) = [1]

The next rules are strictly ordered:

  p1

We remove them from the problem.  Then no dependency pair remains.

-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: first#(mark(X1),X2) -> first#(X1,X2)
p2: first#(X1,active(X2)) -> first#(X1,X2)
p3: first#(active(X1),X2) -> first#(X1,X2)
p4: first#(X1,mark(X2)) -> first#(X1,X2)

and R consists of:

r1: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
r2: active(sqr(|0|())) -> mark(|0|())
r3: active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
r4: active(dbl(|0|())) -> mark(|0|())
r5: active(dbl(s(X))) -> mark(s(s(dbl(X))))
r6: active(add(|0|(),X)) -> mark(X)
r7: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r8: active(first(|0|(),X)) -> mark(nil())
r9: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r10: mark(terms(X)) -> active(terms(mark(X)))
r11: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r12: mark(recip(X)) -> active(recip(mark(X)))
r13: mark(sqr(X)) -> active(sqr(mark(X)))
r14: mark(s(X)) -> active(s(X))
r15: mark(|0|()) -> active(|0|())
r16: mark(add(X1,X2)) -> active(add(mark(X1),mark(X2)))
r17: mark(dbl(X)) -> active(dbl(mark(X)))
r18: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r19: mark(nil()) -> active(nil())
r20: terms(mark(X)) -> terms(X)
r21: terms(active(X)) -> terms(X)
r22: cons(mark(X1),X2) -> cons(X1,X2)
r23: cons(X1,mark(X2)) -> cons(X1,X2)
r24: cons(active(X1),X2) -> cons(X1,X2)
r25: cons(X1,active(X2)) -> cons(X1,X2)
r26: recip(mark(X)) -> recip(X)
r27: recip(active(X)) -> recip(X)
r28: sqr(mark(X)) -> sqr(X)
r29: sqr(active(X)) -> sqr(X)
r30: s(mark(X)) -> s(X)
r31: s(active(X)) -> s(X)
r32: add(mark(X1),X2) -> add(X1,X2)
r33: add(X1,mark(X2)) -> add(X1,X2)
r34: add(active(X1),X2) -> add(X1,X2)
r35: add(X1,active(X2)) -> add(X1,X2)
r36: dbl(mark(X)) -> dbl(X)
r37: dbl(active(X)) -> dbl(X)
r38: first(mark(X1),X2) -> first(X1,X2)
r39: first(X1,mark(X2)) -> first(X1,X2)
r40: first(active(X1),X2) -> first(X1,X2)
r41: first(X1,active(X2)) -> first(X1,X2)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        first#_A(x1,x2) = max{0, x2 - 2}
        mark_A(x1) = x1
        active_A(x1) = max{3, x1 + 1}
  
    precedence:
    
      first# = mark = active
    
    partial status:
  
      pi(first#) = []
      pi(mark) = [1]
      pi(active) = [1]

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: first#(mark(X1),X2) -> first#(X1,X2)
p2: first#(active(X1),X2) -> first#(X1,X2)
p3: first#(X1,mark(X2)) -> first#(X1,X2)

and R consists of:

r1: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
r2: active(sqr(|0|())) -> mark(|0|())
r3: active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
r4: active(dbl(|0|())) -> mark(|0|())
r5: active(dbl(s(X))) -> mark(s(s(dbl(X))))
r6: active(add(|0|(),X)) -> mark(X)
r7: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r8: active(first(|0|(),X)) -> mark(nil())
r9: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r10: mark(terms(X)) -> active(terms(mark(X)))
r11: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r12: mark(recip(X)) -> active(recip(mark(X)))
r13: mark(sqr(X)) -> active(sqr(mark(X)))
r14: mark(s(X)) -> active(s(X))
r15: mark(|0|()) -> active(|0|())
r16: mark(add(X1,X2)) -> active(add(mark(X1),mark(X2)))
r17: mark(dbl(X)) -> active(dbl(mark(X)))
r18: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r19: mark(nil()) -> active(nil())
r20: terms(mark(X)) -> terms(X)
r21: terms(active(X)) -> terms(X)
r22: cons(mark(X1),X2) -> cons(X1,X2)
r23: cons(X1,mark(X2)) -> cons(X1,X2)
r24: cons(active(X1),X2) -> cons(X1,X2)
r25: cons(X1,active(X2)) -> cons(X1,X2)
r26: recip(mark(X)) -> recip(X)
r27: recip(active(X)) -> recip(X)
r28: sqr(mark(X)) -> sqr(X)
r29: sqr(active(X)) -> sqr(X)
r30: s(mark(X)) -> s(X)
r31: s(active(X)) -> s(X)
r32: add(mark(X1),X2) -> add(X1,X2)
r33: add(X1,mark(X2)) -> add(X1,X2)
r34: add(active(X1),X2) -> add(X1,X2)
r35: add(X1,active(X2)) -> add(X1,X2)
r36: dbl(mark(X)) -> dbl(X)
r37: dbl(active(X)) -> dbl(X)
r38: first(mark(X1),X2) -> first(X1,X2)
r39: first(X1,mark(X2)) -> first(X1,X2)
r40: first(active(X1),X2) -> first(X1,X2)
r41: first(X1,active(X2)) -> first(X1,X2)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: first#(mark(X1),X2) -> first#(X1,X2)
p2: first#(X1,mark(X2)) -> first#(X1,X2)
p3: first#(active(X1),X2) -> first#(X1,X2)

and R consists of:

r1: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
r2: active(sqr(|0|())) -> mark(|0|())
r3: active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
r4: active(dbl(|0|())) -> mark(|0|())
r5: active(dbl(s(X))) -> mark(s(s(dbl(X))))
r6: active(add(|0|(),X)) -> mark(X)
r7: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r8: active(first(|0|(),X)) -> mark(nil())
r9: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r10: mark(terms(X)) -> active(terms(mark(X)))
r11: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r12: mark(recip(X)) -> active(recip(mark(X)))
r13: mark(sqr(X)) -> active(sqr(mark(X)))
r14: mark(s(X)) -> active(s(X))
r15: mark(|0|()) -> active(|0|())
r16: mark(add(X1,X2)) -> active(add(mark(X1),mark(X2)))
r17: mark(dbl(X)) -> active(dbl(mark(X)))
r18: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r19: mark(nil()) -> active(nil())
r20: terms(mark(X)) -> terms(X)
r21: terms(active(X)) -> terms(X)
r22: cons(mark(X1),X2) -> cons(X1,X2)
r23: cons(X1,mark(X2)) -> cons(X1,X2)
r24: cons(active(X1),X2) -> cons(X1,X2)
r25: cons(X1,active(X2)) -> cons(X1,X2)
r26: recip(mark(X)) -> recip(X)
r27: recip(active(X)) -> recip(X)
r28: sqr(mark(X)) -> sqr(X)
r29: sqr(active(X)) -> sqr(X)
r30: s(mark(X)) -> s(X)
r31: s(active(X)) -> s(X)
r32: add(mark(X1),X2) -> add(X1,X2)
r33: add(X1,mark(X2)) -> add(X1,X2)
r34: add(active(X1),X2) -> add(X1,X2)
r35: add(X1,active(X2)) -> add(X1,X2)
r36: dbl(mark(X)) -> dbl(X)
r37: dbl(active(X)) -> dbl(X)
r38: first(mark(X1),X2) -> first(X1,X2)
r39: first(X1,mark(X2)) -> first(X1,X2)
r40: first(active(X1),X2) -> first(X1,X2)
r41: first(X1,active(X2)) -> first(X1,X2)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        first#_A(x1,x2) = max{0, x1 - 2}
        mark_A(x1) = max{3, x1 + 1}
        active_A(x1) = x1
  
    precedence:
    
      first# = mark = active
    
    partial status:
  
      pi(first#) = []
      pi(mark) = [1]
      pi(active) = [1]

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: first#(X1,mark(X2)) -> first#(X1,X2)
p2: first#(active(X1),X2) -> first#(X1,X2)

and R consists of:

r1: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
r2: active(sqr(|0|())) -> mark(|0|())
r3: active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
r4: active(dbl(|0|())) -> mark(|0|())
r5: active(dbl(s(X))) -> mark(s(s(dbl(X))))
r6: active(add(|0|(),X)) -> mark(X)
r7: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r8: active(first(|0|(),X)) -> mark(nil())
r9: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r10: mark(terms(X)) -> active(terms(mark(X)))
r11: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r12: mark(recip(X)) -> active(recip(mark(X)))
r13: mark(sqr(X)) -> active(sqr(mark(X)))
r14: mark(s(X)) -> active(s(X))
r15: mark(|0|()) -> active(|0|())
r16: mark(add(X1,X2)) -> active(add(mark(X1),mark(X2)))
r17: mark(dbl(X)) -> active(dbl(mark(X)))
r18: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r19: mark(nil()) -> active(nil())
r20: terms(mark(X)) -> terms(X)
r21: terms(active(X)) -> terms(X)
r22: cons(mark(X1),X2) -> cons(X1,X2)
r23: cons(X1,mark(X2)) -> cons(X1,X2)
r24: cons(active(X1),X2) -> cons(X1,X2)
r25: cons(X1,active(X2)) -> cons(X1,X2)
r26: recip(mark(X)) -> recip(X)
r27: recip(active(X)) -> recip(X)
r28: sqr(mark(X)) -> sqr(X)
r29: sqr(active(X)) -> sqr(X)
r30: s(mark(X)) -> s(X)
r31: s(active(X)) -> s(X)
r32: add(mark(X1),X2) -> add(X1,X2)
r33: add(X1,mark(X2)) -> add(X1,X2)
r34: add(active(X1),X2) -> add(X1,X2)
r35: add(X1,active(X2)) -> add(X1,X2)
r36: dbl(mark(X)) -> dbl(X)
r37: dbl(active(X)) -> dbl(X)
r38: first(mark(X1),X2) -> first(X1,X2)
r39: first(X1,mark(X2)) -> first(X1,X2)
r40: first(active(X1),X2) -> first(X1,X2)
r41: first(X1,active(X2)) -> first(X1,X2)

The estimated dependency graph contains the following SCCs:

  {p1, p2}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: first#(X1,mark(X2)) -> first#(X1,X2)
p2: first#(active(X1),X2) -> first#(X1,X2)

and R consists of:

r1: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
r2: active(sqr(|0|())) -> mark(|0|())
r3: active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
r4: active(dbl(|0|())) -> mark(|0|())
r5: active(dbl(s(X))) -> mark(s(s(dbl(X))))
r6: active(add(|0|(),X)) -> mark(X)
r7: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r8: active(first(|0|(),X)) -> mark(nil())
r9: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r10: mark(terms(X)) -> active(terms(mark(X)))
r11: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r12: mark(recip(X)) -> active(recip(mark(X)))
r13: mark(sqr(X)) -> active(sqr(mark(X)))
r14: mark(s(X)) -> active(s(X))
r15: mark(|0|()) -> active(|0|())
r16: mark(add(X1,X2)) -> active(add(mark(X1),mark(X2)))
r17: mark(dbl(X)) -> active(dbl(mark(X)))
r18: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r19: mark(nil()) -> active(nil())
r20: terms(mark(X)) -> terms(X)
r21: terms(active(X)) -> terms(X)
r22: cons(mark(X1),X2) -> cons(X1,X2)
r23: cons(X1,mark(X2)) -> cons(X1,X2)
r24: cons(active(X1),X2) -> cons(X1,X2)
r25: cons(X1,active(X2)) -> cons(X1,X2)
r26: recip(mark(X)) -> recip(X)
r27: recip(active(X)) -> recip(X)
r28: sqr(mark(X)) -> sqr(X)
r29: sqr(active(X)) -> sqr(X)
r30: s(mark(X)) -> s(X)
r31: s(active(X)) -> s(X)
r32: add(mark(X1),X2) -> add(X1,X2)
r33: add(X1,mark(X2)) -> add(X1,X2)
r34: add(active(X1),X2) -> add(X1,X2)
r35: add(X1,active(X2)) -> add(X1,X2)
r36: dbl(mark(X)) -> dbl(X)
r37: dbl(active(X)) -> dbl(X)
r38: first(mark(X1),X2) -> first(X1,X2)
r39: first(X1,mark(X2)) -> first(X1,X2)
r40: first(active(X1),X2) -> first(X1,X2)
r41: first(X1,active(X2)) -> first(X1,X2)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        first#_A(x1,x2) = x2 + 1
        mark_A(x1) = x1 + 1
        active_A(x1) = x1 + 1
  
    precedence:
    
      first# = mark = active
    
    partial status:
  
      pi(first#) = []
      pi(mark) = [1]
      pi(active) = [1]

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: first#(active(X1),X2) -> first#(X1,X2)

and R consists of:

r1: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
r2: active(sqr(|0|())) -> mark(|0|())
r3: active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
r4: active(dbl(|0|())) -> mark(|0|())
r5: active(dbl(s(X))) -> mark(s(s(dbl(X))))
r6: active(add(|0|(),X)) -> mark(X)
r7: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r8: active(first(|0|(),X)) -> mark(nil())
r9: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r10: mark(terms(X)) -> active(terms(mark(X)))
r11: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r12: mark(recip(X)) -> active(recip(mark(X)))
r13: mark(sqr(X)) -> active(sqr(mark(X)))
r14: mark(s(X)) -> active(s(X))
r15: mark(|0|()) -> active(|0|())
r16: mark(add(X1,X2)) -> active(add(mark(X1),mark(X2)))
r17: mark(dbl(X)) -> active(dbl(mark(X)))
r18: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r19: mark(nil()) -> active(nil())
r20: terms(mark(X)) -> terms(X)
r21: terms(active(X)) -> terms(X)
r22: cons(mark(X1),X2) -> cons(X1,X2)
r23: cons(X1,mark(X2)) -> cons(X1,X2)
r24: cons(active(X1),X2) -> cons(X1,X2)
r25: cons(X1,active(X2)) -> cons(X1,X2)
r26: recip(mark(X)) -> recip(X)
r27: recip(active(X)) -> recip(X)
r28: sqr(mark(X)) -> sqr(X)
r29: sqr(active(X)) -> sqr(X)
r30: s(mark(X)) -> s(X)
r31: s(active(X)) -> s(X)
r32: add(mark(X1),X2) -> add(X1,X2)
r33: add(X1,mark(X2)) -> add(X1,X2)
r34: add(active(X1),X2) -> add(X1,X2)
r35: add(X1,active(X2)) -> add(X1,X2)
r36: dbl(mark(X)) -> dbl(X)
r37: dbl(active(X)) -> dbl(X)
r38: first(mark(X1),X2) -> first(X1,X2)
r39: first(X1,mark(X2)) -> first(X1,X2)
r40: first(active(X1),X2) -> first(X1,X2)
r41: first(X1,active(X2)) -> first(X1,X2)

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: first#(active(X1),X2) -> first#(X1,X2)

and R consists of:

r1: active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
r2: active(sqr(|0|())) -> mark(|0|())
r3: active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
r4: active(dbl(|0|())) -> mark(|0|())
r5: active(dbl(s(X))) -> mark(s(s(dbl(X))))
r6: active(add(|0|(),X)) -> mark(X)
r7: active(add(s(X),Y)) -> mark(s(add(X,Y)))
r8: active(first(|0|(),X)) -> mark(nil())
r9: active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
r10: mark(terms(X)) -> active(terms(mark(X)))
r11: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r12: mark(recip(X)) -> active(recip(mark(X)))
r13: mark(sqr(X)) -> active(sqr(mark(X)))
r14: mark(s(X)) -> active(s(X))
r15: mark(|0|()) -> active(|0|())
r16: mark(add(X1,X2)) -> active(add(mark(X1),mark(X2)))
r17: mark(dbl(X)) -> active(dbl(mark(X)))
r18: mark(first(X1,X2)) -> active(first(mark(X1),mark(X2)))
r19: mark(nil()) -> active(nil())
r20: terms(mark(X)) -> terms(X)
r21: terms(active(X)) -> terms(X)
r22: cons(mark(X1),X2) -> cons(X1,X2)
r23: cons(X1,mark(X2)) -> cons(X1,X2)
r24: cons(active(X1),X2) -> cons(X1,X2)
r25: cons(X1,active(X2)) -> cons(X1,X2)
r26: recip(mark(X)) -> recip(X)
r27: recip(active(X)) -> recip(X)
r28: sqr(mark(X)) -> sqr(X)
r29: sqr(active(X)) -> sqr(X)
r30: s(mark(X)) -> s(X)
r31: s(active(X)) -> s(X)
r32: add(mark(X1),X2) -> add(X1,X2)
r33: add(X1,mark(X2)) -> add(X1,X2)
r34: add(active(X1),X2) -> add(X1,X2)
r35: add(X1,active(X2)) -> add(X1,X2)
r36: dbl(mark(X)) -> dbl(X)
r37: dbl(active(X)) -> dbl(X)
r38: first(mark(X1),X2) -> first(X1,X2)
r39: first(X1,mark(X2)) -> first(X1,X2)
r40: first(active(X1),X2) -> first(X1,X2)
r41: first(X1,active(X2)) -> first(X1,X2)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        first#_A(x1,x2) = max{1, x1, x2}
        active_A(x1) = x1 + 1
  
    precedence:
    
      first# = active
    
    partial status:
  
      pi(first#) = [1, 2]
      pi(active) = []

The next rules are strictly ordered:

  p1

We remove them from the problem.  Then no dependency pair remains.